We then apply Lemma 10.1 to each of these summands. Hence we arrive at

$m_{1}=(1+o_{k_{4}})k_{1}k_{5}e^{-(T_{+}-T_{-})}\frac{c(k_{7},\infty)V_{j}^{\prime}\sqrt{k_{4}}}{(T_{+}-T_{-})^{3/2}}e^{-\sqrt{2}V_{j}^{\prime}-(V_{j}^{\prime})^{2}/2(T_{+}-T_{-})},$

which simplifies to the claimed result. ∎

###### Proof of Proposition 2.23.

We shall use the Poisson process machinery stated in Theorem A.1, which we
shall translate into this context. Due to the nature of the Poisson
approximation, we shall use the nonatomicity of the derivative martingale
proved in Theorem 1.6. We let $\eta>0$ be a parameter which shall be taken to
$0$ after $k_{2}$ to establish the convergence in probability.

##### Step 1: restricting the measures.

As the measure $\mathscr{D}_{\infty}$ is finite and nonatomic, we have that
there is $m\in{\mathbb{N}}$ sufficiently large that

$\mathbb{P}(\max_{j=1,\dots,m}\mathscr{D}_{\infty}([\tfrac{2\pi(j-1)}{m},\tfrac{2\pi(j+1)}{m}])>\eta)\leq\eta,$

where for the case $j=m$, we consider the arc as part of the torus. Hence for
all $k_{2}$ sufficiently large, we have

$\mathbb{P}(\max_{j=1,\dots,m}\mathscr{D}_{k_{2}}([\tfrac{2\pi(j-1)}{m},\tfrac{2\pi(j+1)}{m}])>\eta)\leq
2\eta.$

On the event $\mathcal{E}$ that there exists a $j=1,\dots,m$ such that

$\mathscr{D}_{\infty}([\tfrac{2\pi(j-1)}{m},\tfrac{2\pi(j+1)}{m}]^{c})\leq\eta,$

we conclude

$\mathbb{P}(\mathcal{E}^{c}\cap\\{\mathscr{D}_{\infty}([0,2\pi])\leq
2\eta\\})\geq 1-\eta.$

As we can relate $\mathscr{D}_{\infty}([0,2\pi])$ to intensity of both point
processes, this in effect is saying that neither point process has any points.
Indeed, for either process $\Xi=\operatorname{Extre}_{n}$ or
$\Pi(\mathfrak{m})$

$\partial_{2}(\Xi\cap\Gamma_{k_{7}},0)\leq\mathbb{P}(\Xi(\Gamma_{k_{7}})\geq
1~{}|~{}\mathscr{F}_{n_{1}^{+}})\leq\mathfrak{m}(\Gamma_{k_{7}}^{+}).$

Using Lemma 4.4 and using Corollary 2.24,

$\mathbb{E}(\mathfrak{m}(\Gamma_{k_{7}}^{+}~{}|~{}\mathscr{F}_{k_{2}}))\leq\frac{c(k_{4})}{n_{1}}\sum_{j\in\mathcal{D}_{n/k_{1}}}e^{\sqrt{\tfrac{\beta}{2}}(\varphi_{k_{2}}(\theta_{j}))-\log(k_{2})}\bigl{(}\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{)}_{+}$

On taking $n\to\infty$, we conclude

$\limsup_{n\to\infty}\mathbb{E}(\mathfrak{m}(\Gamma_{k_{7}}^{+}~{}|~{}\mathscr{F}_{k_{2}}))\leq
c(k_{4})\mathscr{D}_{k_{2}}([0,2\pi]).$

And hence on $\mathcal{E}^{c},$ we conclude

$\limsup_{k_{2},k_{1},n\to\infty}\mathbb{P}(\mathcal{E}^{c}\cap\\{\partial_{2}(\operatorname{Extre}_{n}\cap\Gamma_{k_{7}},\Pi(\mathfrak{m}))>4c(k_{4})\eta\\})\leq\eta.$

Hence it suffices to work on the event $\mathcal{E}^{c}$.

##### Step 2: setting up the Poisson approximation.

We let $\delta>0$ be the same constant as in (2.36) (so that Proposition 2.12
applies), and define for $j\in\mathcal{D}_{n/k_{1}}$

$\mathcal{B}_{j}=\left\\{k\in\mathcal{D}_{n/k_{1}}:d_{\mathbb{T}}(\theta_{j},\theta_{k})\leq
n^{-1+\delta}\right\\},\quad{\text{and}}\quad
B_{j}=\left\\{k\in\mathcal{D}_{n/k_{1}}:d_{\mathbb{T}}(\theta_{j},\theta_{k})\leq
3n^{-1+\delta}\right\\}$

with $d_{\mathbb{T}}$ the distance in the quotient space
${\mathbb{R}}/(2\pi{\mathbb{Z}}).$ Recall (2.39) and (2.46). By construction,
for all $j\in\mathcal{D}_{n/k_{1}}$,

$\\{(V_{k}^{\prime},D_{k}^{o}e^{\sqrt{4/\beta}V_{k}^{\prime}}):k\in\mathcal{B}_{j}\\}\quad\text{is
$\bigl{(}\mathscr{F}_{{n}_{1}^{+}}\bigr{)}$--conditional independent
of}\quad\\{(V_{k}^{\prime},D_{k}^{o}e^{-V_{k}^{\prime}}):k{\not\in}B_{j}\\}.$

Define for any $j\in\mathcal{D}_{n/k_{1}},$

$\Xi_{j}\coloneqq\delta_{(\theta_{j},V_{j}^{\prime},D_{j}^{o})}\times\mathbf{1}\left\\{{\mathscr{R}_{j}^{2}(n_{1}^{+})}\right\\},\,P_{j}\coloneqq\Xi_{j}(\Gamma_{k_{7}}^{+}),\,S_{j}\coloneqq\sum_{i\in\mathcal{B}_{j}\setminus\\{j\\}}P_{i},\,\text{and}\quad
L_{j}\coloneqq\sum_{i\in\mathcal{D}_{n/k_{1}}\setminus
B_{j}}\mathfrak{m}_{i}(\Gamma_{k_{7}}^{+}).$

We note that
$\mathbb{E}(P_{i}~{}|~{}\mathscr{F}_{n_{1}^{+}})=\mathfrak{m}_{i}(\Gamma_{k_{7}}^{+}).$
Theorem A.1 shows that there is a numerical constant $C>0$ so that with
$\Pi=\Pi(\mathfrak{q}),$

$\displaystyle\partial_{2}(\operatorname{Extr}_{n}\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{{n}_{1}^{+}},\Pi(\mathfrak{m})\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{{n}_{1}^{+}})$
(2.53) $\displaystyle\leq$
$\displaystyle\partial_{2}(\operatorname{Extr}_{n}\cap\Gamma_{k_{7}}^{+}~{}|~{}\mathscr{F}_{{n}_{1}^{+}},\Pi(\mathfrak{m})\cap\Gamma_{k_{7}}^{+}~{}|~{}\mathscr{F}_{{n}_{1}^{+}})$
$\displaystyle\leq$ $\displaystyle
C[\operatorname{Var}(\operatorname{Extre}_{n}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{n_{1}^{+}})+3\mathfrak{m}(\Gamma_{k_{7}}^{+})]^{3/2}$
$\displaystyle\times\sum_{j\in\mathcal{D}_{n/k_{1}}}\biggl{(}\frac{\mathbb{E}(P_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}})\mathbb{E}(S_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}})+\mathbb{E}(S_{j}P_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}})+(\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+}))^{2}}{L_{j}^{2}}\biggr{)}.$

We also note that the left hand side of (2.53) is bounded by $1,$ and hence it
suffices to bound the right hand side of (2.53) on any
$\bigl{(}\mathscr{F}_{{n}_{1}^{+}}\bigr{)}$–measurable event with probability
tending to $1$. Nonetheless, as input, we need estimates for the
$\bigl{(}\mathscr{F}_{n_{1}^{+}}\bigr{)}$–conditional expectation of $P_{j}$
and for pairs $P_{j}$ and $P_{i}$ where $i\in\mathcal{B}_{j}.$ We note that
$\mathbb{E}[P_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}}]=\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+}).$

To complete the proof of the claim, we show that there are events
$\mathcal{F}=\mathcal{F}(n,(k_{j}))$ so that on
$\mathcal{F}\cap\mathcal{E}^{c}$ the following hold

  1. 1.

$\mathfrak{m}(\Gamma_{k_{7}}^{+})=O_{k_{2}}(1)$,

  2. 2.

$\operatorname{Var}(\operatorname{Extr}_{n}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{n_{1}^{+}})=O_{k_{2}}(1)$,

  3. 3.

$L_{j}^{2}\geq c(k_{4})\eta^{2}$,

and so that
$\mathbf{1}\left\\{{\mathcal{F}}\right\\}\xrightarrow[k_{2},k_{1},n\to\infty]{\mathbb{P}}1$.

##### Step 3: completing the proof on the good event.

Having shown the claim, we have reduced the problem to showing

$\displaystyle\sum_{j\in\mathcal{D}_{n/k_{1}}}\biggl{(}\underbracket{\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+})\mathbb{E}(S_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}})}_{(i)}+\underbracket{\mathbb{E}(S_{j}P_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}})}_{(ii)}+\underbracket{\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+})^{2}}_{(iii)}\biggr{)}\xrightarrow[k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$
(2.54)

For the term $(iii),$ we note that $\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+}),$
which goes to $0$ uniformly in $j$ faster than any power of $\log k_{1}$ (from
Corollary 2.24). Hence by the boundededness of
$\mathfrak{m}(\Gamma_{k_{7}}^{+})$ on $\mathcal{F}$, this term tends to $0.$

For term $(i),$ using the second moment machinery, we arrive at (bounding the
sum on $z_{j},z_{\ell}$ by an integral, which holds up to a constant depending
on $c_{4}$)

$\mathbb{E}[~{}(i)~{}|~{}\mathscr{F}_{k_{2}}]\leq
c(k_{3})\bigl{(}\frac{k_{1}}{k_{1}^{+}}\bigr{)}^{2}\sum_{\ell\in\mathcal{B}_{j}}\int_{z_{j},z_{\ell}}\mathscr{Q}(\theta_{j},x_{j},z_{j};\theta_{\ell},x_{\ell},z_{\ell})\frac{z_{j}z_{\ell}}{(\log
k_{1})^{3}}e^{-\sqrt{2}(z_{j}+z_{\ell})}dz_{j}dz_{\ell}.$ (2.55)

We divide the angles $\theta_{\ell}$ according to whether or not
$(\log_{2}n_{1}^{+}-\mathtt{k}_{+})\geq q$ or
$(\log_{2}n_{1}^{+}-\mathtt{k}_{+})\leq q$ for a $q$ to be chosen later. For
the former case we use the third case of Proposition 2.21. For the latter
case, we just bound
$\mathscr{Q}(\theta_{j},x_{j},z_{j};\theta_{\ell},x_{\ell},z_{\ell})$ by the
probability of the $\theta_{j}$–ray event. We also note that the number of
$\theta_{j}$ in this latter case is $(k_{1}^{+}/k_{1})e^{q+100(\log\log
k_{1})^{2}}$. Applying the bound from this case, we arrive at

$\displaystyle\mathbb{E}[~{}(i)~{}|~{}\mathscr{F}_{k_{2}}]$
$\displaystyle\leq\frac{c(k_{4})}{n_{1}}e^{\sqrt{\tfrac{\beta}{2}}(\varphi_{k_{2}}(\theta_{j}))-\log(k_{2})}\bigl{(}\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{)}$ (2.56)
$\displaystyle\times(\log k_{1})^{3}\biggl{\\{}\sum_{\varkappa=q}^{\delta\log
n}\bigl{\\{}\exp\bigl{(}-c\varkappa^{1/10}\bigr{)}\bigr{\\}}+\exp\bigl{(}q+C(\log\log
k_{1})^{2}-c(\log k_{1})^{1/10}\bigr{)}\biggr{\\}}.$

Note that the stretched exponential gain in the second term is simply from the
entropic envelope and the $e^{-\sqrt{2}V_{\ell}^{\prime}}$ term from Corollary
2.24. As the event we consider restricts the location of
$\varphi_{k_{2}}(\theta_{j})$ to be positive, we may use that

$\sum_{j\in\mathcal{D}_{n/k_{1}}}\frac{1}{n_{1}}e^{\sqrt{\tfrac{\beta}{2}}(\varphi_{k_{2}}(\theta_{j}))-\log(k_{2})}\bigl{(}\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{)}_{+}$

converges on taking $n\to\infty$. Taking $q=(\log k_{1})^{1/100}$, the sum is
on the order of $e^{-\Omega((\log k_{1})^{1/1000})}$. Hence on taking
$k_{1}\to\infty$,

$\sum_{j\in\mathcal{D}_{n/k_{1}}}\mathbb{E}[~{}(i)~{}|~{}\mathscr{F}_{k_{2}}]\xrightarrow[k_{1},n\to\infty]{\mathbb{P}}0.$

The analysis for $\mathbb{E}[~{}(ii)~{}|~{}\mathscr{F}_{k_{2}}]$ is similar,
but with one important modification. For bushes $\theta_{j}$ and
$\theta_{\ell}$ that branch at or before $n_{1}^{+}$ (i.e.
$\mathtt{k}\leq\log_{2}n_{1}^{+}$) we claim that the bound in (2.55) holds as
well: we bound each indicator
$\mathbf{1}\left\\{{W_{j}^{o}\in[-k_{7},\infty)}\right\\}$ above by a sum of
indicators of ray event and then use Lemma 10.1. For $\theta_{\ell}$ which are
close to $\theta_{j}$, we use the same strategy, although we instead lose the
precise dependence on $z_{\ell}$ and instead have just the factor
$e^{-\Omega(\log k_{1})^{1/10}}.$ For $\ell$ so that
$\mathtt{k}\geq\log_{2}n_{1}^{+}$, this produces the bound:

$\sum_{j\in\mathcal{D}_{n/k_{1}}}c(k_{3})\bigl{(}\frac{k_{1}}{k_{1}^{+}}\bigr{)}^{2}\sum_{\ell}\mathscr{Q}(\theta_{j},x_{j},z_{j};\theta_{\ell},x_{\ell},z_{\ell})\frac{z_{j}}{(\log
k_{1})^{3/2}}e^{-\sqrt{2}(z_{j})-c(\log k_{1})^{1/10}}.$

Using the final case of Proposition 2.21 gives an estimate which is
$e^{-\Omega((\log k_{1})^{1/10})}$.

##### Step 4: Proof of claim.

_Point 1._ We have that

$\mathfrak{m}(\Gamma_{k_{7}}^{+})=\sum_{j\in\mathcal{D}_{n/k_{1}}}\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+}),$

and so using Corollary (2.24) and Lemma 4.4

$\mathbb{E}[\mathfrak{m}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{k_{2}}]\leq\frac{c(k_{4})}{n_{1}}\sum_{j\in\mathcal{D}_{n/k_{1}}}e^{\sqrt{\tfrac{\beta}{2}}(\varphi_{k_{2}}(\theta_{j}))-\log(k_{2})}\bigl{(}\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{)}_{+}.$

This converges on taking $n\to\infty$ almost surely to

$\limsup_{n\to\infty}\mathbb{E}[\mathfrak{m}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{k_{2}}]\leq
c(k_{4})\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta,$ (2.57)

and so remains bounded almost surely by Theorem 1.6.

_Point 2._ It suffices to bound the conditional second moment, which is to say

$\operatorname{Var}(\operatorname{Extre}_{n}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{n_{1}^{+}})\leq\mathbb{E}[\operatorname{Extre}_{n}^{2}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{n_{1}^{+}}].$

We then have

$\mathbb{E}[\operatorname{Extre}_{n}^{2}(\Gamma_{k_{7}}^{+})~{}|~{}\mathscr{F}_{n_{1}^{+}}]\leq\sum_{j\in\mathcal{D}_{n/k_{1}}}\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+})+\sum_{j\in\mathcal{D}_{n/k_{1}}}\mathbb{E}[P_{j}S_{j}~{}|~{}\mathscr{F}_{n_{1}^{+}}]+\sum_{j\in\mathcal{D}_{n/k_{1}}}\sum_{\ell\not\in\mathcal{B}_{j}}\mathfrak{m}_{j}(\Gamma_{k_{7}}^{+})\mathfrak{m}_{\ell}(\Gamma_{k_{7}}^{+})$

The first term is nothing but $\mathfrak{m}(\Gamma_{k_{7}}^{+})$ which we have
already controlled. The second term we controlled earlier in parts $(ii)$ and
$(i)$ above. The third term we estimate in the same fashion as $(i)$ in
(2.56).

_Point 3._ We let $\text{Arc}_{i}$ for $i=1,\dots,m$ be a Lipschitz function
of the torus ${\mathbb{R}}/2\pi{\mathbb{Z}}$ which is $1$ on the complement of
$[\tfrac{i-1}{m}2\pi,\tfrac{i+1}{m}2\pi]^{c}$ and $0$ on
$[\tfrac{i-0.5}{m}2\pi,\tfrac{i+0.5}{m}2\pi]$. Extend $\text{Arc}_{i}$ to a
function of $\Gamma_{k_{7}}^{+}$ by setting
$\text{Arc}_{i}(x,y,z)=\text{Arc}_{i}(x).$ Then for each
$j\in\mathcal{D}_{n/k_{1}}$, for all $n$ sufficiently large there is an $i\in
1,\dots,m$ so that $L_{j}\geq\mathfrak{m}(\text{Arc}_{i}).$ Hence it suffices
to show that each of these $\mathfrak{m}(\text{Arc}_{i})$ satisfies the
claimed bound.

Now we claim that in fact $\mathfrak{m}(\text{Arc}_{i})$ concentrates around
its $\mathscr{F}_{k_{2}}$–conditional mean and that its conditional mean has
the claimed lower bound. In fact, we shall need this concentration argument at
a later point as well, and so we formulate a general statement here.

######  Lemma 2.25.

Let $f$ be a non-negative, Lipschitz function from the torus
${\mathbb{R}}/(2\pi{\mathbb{Z}})\to{\mathbb{R}}$ bounded above by $1$. Extend
it to a function of
${\mathbb{R}}/(2\pi{\mathbb{Z}})\times{\mathbb{R}}_{+}\times\mathcal{C}([-2\pi
k_{1},0],{\mathbb{C}})\cap\Gamma_{k_{7}}^{+}$ by setting $f(x,y,z)=f(x)$. Then
there is positive constant $\mathcal{H}=\mathcal{H}(k_{1},k_{4},k_{5},k_{7})$
which is bounded in $k_{1}$ so that

$\mathfrak{m}(f)-\mathscr{D}_{k_{2}}(f)\times\mathcal{H}\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$

The constant $\mathcal{H}$ is given in (2.58).

This completes the proof of the proposition. ∎

###### Proof of Lemma 2.25.

If $\mathscr{D}_{\infty}(f)=0$ then it suffices to show $\mathfrak{m}(f)$
tends to $0$. This follows directly from (2.57), and so it suffices to
consider the case that $\mathscr{D}_{\infty}(f)>0.$

The proof follows from a second moment method, restricted to the event
$\mathscr{Z}_{k_{3}}$, which complicates the first moment estimate. We first
compute the first moment without the restriction to $\mathscr{Z}_{k_{3}}$. The
conditional first moment of $\mathfrak{m}(f)$ is given by

$\mathbb{E}[\mathfrak{m}(f)~{}|~{}\mathscr{F}_{k_{2}}]=\sum_{j\in\mathcal{D}_{n/k_{1}}}f(\theta_{j})\mathbb{E}[~{}\mathscr{R}_{j}^{2}(n_{1}^{+})p_{j}~{}|~{}\mathscr{F}_{k_{2}}]$

where
$p_{j}=p(V_{j}^{\prime})=\mathbb{P}(W_{j}^{o}\in[-k_{7},\infty)~{}|~{}\mathscr{F}_{n_{1}^{+}}).$
The function $p$ is inexplicit and depends on $k_{4},k_{5},k_{7}$. Using Lemma
4.4,

$\mathbb{E}[~{}\mathscr{R}_{j}^{2}(n_{1}^{+})p_{j}~{}|~{}\mathscr{F}_{k_{2}}]\leq\frac{1}{n_{1}}\bigl{(}\sqrt{2}H_{k_{2}}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta_{j})\bigr{)}e^{\sqrt{\beta/2}\varphi_{k_{2}}(\theta_{j})-\log
k_{2}+o_{k_{2}}}\sqrt{\frac{2}{\pi}}\int_{J}\frac{k_{1}^{+}}{k_{1}}e^{\sqrt{2}z}p(z)z\,dz$

The interval $J$ is $[(\log k_{1}^{+})^{1/10},(\log k_{1}^{+})^{9/10}]$. The
constant $\mathcal{H}$ is given by

$\mathcal{H}\coloneqq\int_{J}\biggl{(}\sqrt{\frac{2}{\pi}}ze^{\sqrt{2}z}\biggr{)}\times\biggl{(}\frac{k_{1}^{+}}{k_{1}}p(z)\biggr{)}\,dz.$
(2.58)

Note that the density that appears is bounded using Corollary 2.24 (using
$T_{-}=\log(k_{1}^{+}/k_{1})$ and $T_{+}=\log k_{1}$), and we obtain

$\mathcal{H}\leq c(k_{4})\int_{J}e^{-z^{2}/(2\log
k_{1}^{+})}\frac{z^{2}dz}{(\log k_{1}^{+})^{3/2}}\leq c^{\prime}(k_{4}).$
(2.59)

We return to this bound in a moment, but note that by combining the above, we
have the upper bound for the conditional first moment:

$\limsup_{n\to\infty}\mathbb{E}[\mathfrak{m}(f)~{}|~{}\mathscr{F}_{k_{2}}]\leq\int_{0}^{2\pi}f(\theta)\mathscr{D}_{k_{2}}(\theta)d\theta\times\mathcal{H}.$
(2.60)

To produce a lower bound for the first moment, we introduce two comparison
measures $\mathfrak{m}_{j}^{\prime}$ and $\mathfrak{m}_{j}^{\prime\prime}$,
which serve as comparisons to $\mathfrak{m}_{j}$, and which are modified by
slightly adjusting the ray event $\mathscr{R}_{j}^{2}(n_{1}^{+})$. We
introduce two ray events, which will only be used for this argument,

$\displaystyle\mathscr{V}_{j}^{\prime}=\mathscr{U}(\theta_{j})\bigcap\\{\forall~{}t\in[H_{k_{2}},H_{n_{1}^{+}}]\cap\\{H_{2^{k}}:k\in{\mathbb{N}}\\},~{}:~{}\mathfrak{Z}_{t}(\theta_{j})\in\sqrt{2}[A_{t}^{5/2,-},A_{t}^{5/2,+}]\\}$
(2.61) $\displaystyle\qquad\qquad\bigcap\\{-V_{j}^{\prime}\in[(\log
k_{1})^{0.49},(\log k_{1})^{0.51}\\},\quad\text{ and }$
$\displaystyle\mathscr{V}_{j}^{\prime\prime}=\mathscr{U}(\theta_{j})\bigcap\\{\forall~{}t\in[H_{k_{2}},H_{n_{1}^{+}}]\cap\\{H_{2^{k}}:k\in{\mathbb{N}}\\},~{}:~{}Z^{k_{2}}_{2^{n}}(\theta_{j})\in\sqrt{2}[A_{t}^{2,-},A_{t}^{2,+}]\\}$
$\displaystyle\qquad\qquad\bigcap\\{-V_{j}^{\prime}\in[(\log
k_{1})^{0.49},(\log k_{1})^{0.51}\\}.$

Let $\mathfrak{m}^{\prime}$ and $\mathfrak{m}^{\prime\prime}$ be the sum of
all the $\mathfrak{m}_{j}^{\prime}$ and $\mathfrak{m}_{j}^{\prime\prime}$
respectively (see (2.50)). From Lemma 4.4 and a direct first moment estimate

$\liminf_{n\to\infty}\mathbb{E}[|\mathfrak{m}^{\prime}(f)-\mathfrak{m}(f)|~{}|~{}\mathscr{F}_{k_{2}}]\leq
o_{k_{2}}\mathscr{D}_{k_{2}}(f),\quad\operatorname{a.s.}$

and hence on taking $k_{2}\to\infty,$ this tends to $0$ almost surely.
Moreover, on the event $\mathscr{Z}_{k_{3}}$, we have that
$\mathfrak{m}^{\prime}\geq\mathfrak{m}^{\prime\prime}$, and hence combining
this with the display above

$\liminf_{n\to\infty}\mathbb{E}[\mathfrak{m}(f)~{}|~{}\mathscr{F}_{k_{2}}]\geq\liminf_{n\to\infty}\mathbb{E}[\mathfrak{m}^{\prime\prime}(f)\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}~{}|~{}\mathscr{F}_{k_{2}}].$

To evaluate this expectation, we lower bound the expectation without the
restriction to the event $\mathscr{Z}_{k_{3}}$ and then argue the event can be
removed.

Using Lemma C.1 and the Girsanov transformation, we also can give a sharp
estimate for the ray probability. However, this is not given as a density
estimate, but rather with a restriction of the endpoint $V_{j}^{\prime}$ to
land in an interval of length $1/\sqrt{k_{3}}$ (as in Proposition 2.21). Hence
we partition the interval $J$ into a grid of separation $1/\sqrt{k_{3}}$.

$\displaystyle\mathbb{E}[\mathfrak{m}^{\prime\prime}(f)~{}|~{}\mathscr{F}_{k_{2}}]=$
$\displaystyle\sum_{j\in\mathcal{D}_{n/k_{1}}}\mathbb{E}\bigl{(}\mathbf{1}\left\\{{\mathscr{V}_{j}^{\prime\prime}}\right\\}p_{j}~{}\big{|}~{}\mathscr{F}_{k_{2}}\bigr{)}$
$\displaystyle\geq$
$\displaystyle\sum_{j\in\mathcal{D}_{n/k_{1}}}\frac{1}{n_{1}}\bigl{(}\sqrt{2}H_{k_{2}}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta_{j})\bigr{)}e^{\sqrt{\beta/2}\varphi_{k_{2}}(\theta_{j})-\log
k_{2}+o_{k_{2}}}$ $\displaystyle\hskip
30.00005pt\times(1+\eta_{k_{2},k_{3}})\frac{k_{1}^{+}}{k_{1}}\sum_{z}\sqrt{\frac{2}{\pi}}ze^{\sqrt{2}z}\times\min_{x\in[0,1/\sqrt{k_{3}}]}\frac{p(z+x)}{\sqrt{k_{3}}},$

where the factor $\eta_{k_{2},k_{3}}\to 0$ as $k_{2}\to\infty$ followed by
$k_{3}\to\infty$. In Lemma 10.2 we show that the function $p(z)$ satisfies the
estimate

$p(z_{j}+x)=(e^{\sqrt{2}x}+o_{k_{1}})p(z_{j})\quad\text{uniformly over
$|x|\leq 1$ and $z_{j}$ in $J$. }$ (2.62)

Hence, we can uniformly approximate the sum over $z$ by an integral and arrive
at

$\displaystyle\mathbb{E}[\mathfrak{m}^{\prime\prime}(f)~{}|~{}\mathscr{F}_{k_{2}}]$
$\displaystyle\geq(1+o_{k_{3}})\sum_{j\in\mathcal{D}_{n/k_{1}}}\frac{1}{n_{1}}\bigl{(}\sqrt{2}H_{k_{2}}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta_{j})\bigr{)}e^{\sqrt{\beta/2}\varphi_{k_{2}}(\theta_{j})-\log
k_{2}}\frac{k_{1}^{+}}{k_{1}}\int_{J}\sqrt{\frac{2}{\pi}}ze^{\sqrt{2}z}p(z)\,dz.$

This is exactly $\mathcal{H}$, and so we have

$\liminf_{n\to\infty}\mathbb{E}[\mathfrak{m}^{\prime\prime}(f)~{}|~{}\mathscr{F}_{k_{2}}]\geq(1+o_{k_{3}})\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta\times\mathcal{H}.$
(2.63)

To complete the first moment analysis, it suffices to show that

$\limsup_{k_{2}\to\infty}\limsup_{k_{1},n\to\infty}\mathbb{E}[\mathfrak{m}^{\prime\prime}(f)\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}^{c}}\right\\}~{}|~{}\mathscr{F}_{k_{2}}]\xrightarrow[k_{3}\to\infty]{\mathbb{P}}0,$

as then from (2.63) and (2.60),

$\mathbb{E}[\mathfrak{m}(f)\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}~{}|~{}\mathscr{F}_{k_{2}}]-\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta\times\mathcal{H}\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$
(2.64)

We shall show using the second moment machinery in Proposition 2.21 (and see
Remark 2.22), that

$\limsup_{k_{2}\to\infty}\limsup_{k_{1},n\to\infty}\mathbb{E}[\bigl{(}\mathfrak{m}^{\prime\prime}(\text{Arc}_{i})\bigr{)}^{2}~{}|~{}\mathscr{F}_{k_{2}}]<\infty\quad\operatorname{a.s.}$
(2.65)

And hence it follows by Cauchy–Schwarz and Lemma 2.20

$\limsup_{k_{2}\to\infty}\limsup_{k_{1},n\to\infty}\mathbb{E}[\mathfrak{m}^{\prime\prime}(\text{Arc}_{i})\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}^{c}}\right\\}~{}|~{}\mathscr{F}_{k_{2}}]\xrightarrow[k_{3}\to\infty]{\mathbb{P}}0.$

We turn to the conditional second moment, and note that this argument will
give the same upper bound as was needed for the second moment of
$\mathfrak{m}^{\prime\prime}$ in (2.65) (see Remark 2.22):

$\mathbb{E}[\mathfrak{m}(f)^{2}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}~{}|~{}\mathscr{F}_{k_{2}}]\leq\sum_{j,\ell}f(\theta_{j})f(\theta_{\ell})\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j}^{2}(n_{1}^{+})\cap\mathscr{R}_{\ell}^{2}(n_{1}^{+})}\right\\}p_{j}p_{\ell}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)},$
(2.66)

where
$p_{j}=p(V_{j}^{\prime})=\mathbb{P}(W_{j}^{o}\in[-k_{7},\infty)~{}|~{}\mathscr{F}_{n_{1}^{+}})$
and the sum is over all $j,\ell\in\mathcal{D}_{n/k_{1}}.$ We divide the angle
pairs $(\theta_{j},\theta_{\ell})$ into two classes $\operatorname{NR}$ and
$\operatorname{FR}$, where the latter means $\theta_{j}$ and $\theta_{\ell}$
have branchpoint $\mathtt{k}$ (in the notation of Proposition 2.21) less than
$\tfrac{1}{2}\log_{2}k_{2}$. We let $\operatorname{NR}$ be all other pairs.
For the well–separated angles $\operatorname{FR}$, we can afford only
multiplicative errors of the form $1+o_{k_{3}}$. Now Proposition 2.21 gives
such an estimate, albeit only after asking the endpoint to be in a bin of size
$(1/\sqrt{k_{3}})$.

Thus by partitioning the two-ray expectation in (2.66) according to the values
of $V_{j}^{\prime}$ and $V_{\ell}^{\prime}$ (using bins of size
$(1/\sqrt{k_{3}})$ so to apply Proposition 2.21), we have for
$(\theta_{j},\theta_{\ell})\in\operatorname{FR}$ the contribution to (2.66) is
bounded above by

$\sum_{z_{j},z_{\ell}}\mathscr{Q}(\theta_{j},x_{j},z_{j};\theta_{\ell},x_{\ell},z_{\ell})\times\max_{u_{j},u_{\ell}\leq
1/\sqrt{k_{3}}}p(z_{j}+u_{j})p(z_{\ell}+u_{\ell}).$

By (2.62), the maximum value of $p$ over a short interval of length
$1/\sqrt{k_{3}}$ is its average value over the same interval up to an error
$o_{k_{3}}$. Using the first case of Proposition 2.21, we conclude with that
notation

$\displaystyle\sum_{\theta_{j},\theta_{\ell}\in\operatorname{FR}}\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}$
$\displaystyle\leq(1+\eta_{k_{2},k_{3}}+o_{k_{3}})\sum_{\theta_{j},\theta_{\ell}}\int_{J}\int_{J}\frac{2}{\pi}\frac{z_{j}x_{j}k_{2}}{n_{1}^{+}}\frac{z_{\ell}x_{\ell}k_{2}}{n_{1}^{+}}\exp\bigl{(}\sqrt{2}(z_{j}-x_{j}+z_{\ell}-x_{\ell})\bigr{)}p(z_{1})p(z_{2})dz_{1}dz_{2},$

where the sum is over all $\theta_{j},\theta_{\ell}$ in the arc and $J$ is as
before. Thus on taking $n\to\infty,$

$\displaystyle\limsup_{n\to\infty}\sum_{\theta_{j},\theta_{\ell}\in\operatorname{FR}}$
$\displaystyle\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}$
(2.67)
$\displaystyle\leq(1+o_{k_{3}})\biggl{(}\int_{0}^{2\pi}f(\theta)\mathscr{D}_{k_{2}}(\theta)d\theta\times\mathcal{H}\biggr{)}^{2}.$

For the near terms, we need to use all the cases of the 2-ray bound
Proposition 2.21:

$\displaystyle\sum_{\theta_{j},\theta_{\ell}\in\operatorname{NR}}$
$\displaystyle\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}$
$\displaystyle\leq
c(k_{3})\sum_{\theta_{j},\theta_{\ell}\in\operatorname{NR}}\int_{J}\int_{J}\mathscr{Q}(\theta_{j},x_{j},z_{j};\theta_{\ell},x_{\ell},z_{\ell})p(z_{j})p(z_{\ell})dz_{j}dz_{\ell}.$

We then partition these near terms into which case of Proposition 2.21 the
pairs land. That is we define the sets:

  1. 1.

$\operatorname{NR}_{1}(\theta_{j})$ are all those $\ell$ so that
$\mathtt{k}\geq(\log_{2}k_{2})/2$ but $\mathtt{k}_{+}\leq\log_{2}k_{2}$.

  2. 2.

$\operatorname{NR}_{2}(\theta_{j})$ are all those $\ell$ not in
$\operatorname{NR}_{1}(\theta_{j})$ but so that $\mathtt{k}\leq(\log_{2}n)/2$.

  3. 3.

$\operatorname{NR}_{3}(\theta_{j})$ are all those $\ell$ not in
$\operatorname{NR}_{i}(\theta_{j})$ for $i\in\\{1,2\\}$ but so that
$\mathtt{k}\leq\log_{2}n_{1}^{+}$.

  4. 4.

$\operatorname{NR}_{4}(\theta_{j})$ are all those $\ell$ not in
$\operatorname{NR}_{i}(\theta_{j})$ for $i\in\\{1,2,3\\}$.

In all cases, we have sharp dependence on $x_{j}$ and $z_{j}$, and so we can
integrate over $z_{j}$ to give exactly $\mathcal{H}$ up to an error depending
on $k_{3}$. For the terms in $\operatorname{NR}_{1}(\theta_{j})$, we also have
sharp dependence on $x_{\ell}$ and $z_{\ell}$ up to a multiplicative error,
and hence we have

$\displaystyle\limsup_{n\to\infty}$
$\displaystyle\sum_{j\in\mathcal{D}_{n/k_{1}}}\sum_{\ell\in\operatorname{NR}_{1}(\theta_{j})}\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}$
$\displaystyle\leq
c(k_{3})\biggl{(}\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta\times\mathcal{H}\biggr{)}\times\max_{\theta}\mathscr{D}_{k_{2}}([\theta,\theta+1/\sqrt{k_{2}}]).$

By nonatomicity of $\mathscr{D}_{\infty}$,

$\max_{\theta}\mathscr{D}_{k_{2}}([\theta,\theta+1/\sqrt{k_{2}}])\xrightarrow[k_{2}\to\infty]{\operatorname{a.s.}}0.$

For the terms of the second type, the bound in Proposition 2.21 loses the
dependence in $x_{\ell},$ but we gain a factor due to the entropic barrier,
which is summable

$\displaystyle\limsup_{n\to\infty}$
$\displaystyle\sum_{j\in\mathcal{D}_{n/k_{1}}}\sum_{\ell\in\operatorname{NR}_{2}(\theta_{j})}\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}$
$\displaystyle\leq
c(k_{3})\biggl{(}\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta\times\mathcal{H}\biggr{)}\times
e^{-c(\log k_{2})^{1/10}}.$

The same bound holds for terms of the third type, using the entropic barrier
gain, but now gaining a factor $e^{-c(\log k_{1})^{1/10}}$. Finally for the
terms of fourth type, we use Corollary 2.24, due to which we gain the same
$e^{-c(\log k_{1})^{1/10}}$. In all, we have that the sum over all
$\operatorname{NR}$ pairs satisfies

$\displaystyle\sum_{\theta_{j},\theta_{\ell}\in\operatorname{NR}}$
$\displaystyle\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}\mathbf{1}\left\\{{\mathscr{R}_{j_{1}}^{2}(n_{1}^{+})\cap\mathscr{R}_{j_{2}}^{2}(n_{1}^{+})}\right\\}p_{j_{1}}p_{j_{2}}~{}\bigg{|}~{}\mathscr{F}_{k_{2}}\biggr{)}\leq
o_{k_{2}}\biggl{(}\int_{0}^{2\pi}\mathscr{D}_{k_{2}}(\theta)\,d\theta\times\mathcal{H}\biggr{)}.$

Combining this with (2.67), we have that

$\limsup_{k_{1},n\to\infty}\operatorname{Var}(\mathfrak{m}(f)\mathbf{1}\left\\{{\mathscr{Z}_{k_{3}}}\right\\}~{}|~{}\mathscr{F}_{k_{2}})\xrightarrow[k_{3},k_{2}\to\infty]{\mathbb{P}}0.$

This together with (2.64) proves the lemma. ∎

###  2.9 Third Poisson approximation: concentration of the intensity

The final Poisson approximation replaces the intensity $\mathfrak{m}$ by
(essentially) its $\mathscr{F}_{k_{2}}$–conditional expectation. This is done
by a first and second moment computation. The measure to which we compare
$\mathfrak{m}$ is the one given in the introduction (1.20), namely with $I(v)$
as in (1.20):

$\mathfrak{n}(\theta,v,f)\coloneqq\mathscr{D}_{k_{2}}(\theta)d\theta\otimes
I(v)dv\otimes\mathfrak{p}(df)\quad\text{where}\quad\mathfrak{p}(df)\coloneqq\begin{cases}\int_{0}^{2\pi}\mathfrak{s}_{k_{1},k_{5}}(e^{i\xi}df)\tfrac{d\xi}{2\pi},&\text{
if $\sigma=1$,}\\\ \mathfrak{s}_{k_{1},k_{5}}(df),&\text{ otherwise.
}\end{cases}$ (2.68)

######  Proposition 2.26.

For any $k_{7}$ the restrictions of $\Pi(\mathfrak{m})$ and
$\Pi(\mathfrak{n})$ to $\Gamma_{k_{7}}$ satisfy

$\partial_{2}(\Pi(\mathfrak{m})\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{k_{2}},\Pi(\mathfrak{n})\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{k_{2}})\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$

###### Proof.

In the first step, we replace the measure $\mathfrak{m}$ by one in which the
decoration is averaged, that is, we compare to

$\overline{\mathfrak{m}}(\theta,v,f)\coloneqq\sum_{j\in{\mathcal{D}}_{n/k_{1}}}\mathbf{1}\left\\{{\mathscr{R}^{2}_{j}({n}_{1}^{+})}\right\\}\delta_{\theta_{j},V_{j}^{\prime}}(d\theta,dv)\mathfrak{p}\bigl{(}df\bigr{)}.$
(2.69)

Note that this is identical to $\mathfrak{m}$ in the case $\sigma=i$. The
intensities $\mathfrak{m}(\Gamma_{k_{7}})$ and
$\overline{\mathfrak{m}}(\Gamma_{k_{7}})$ are tight in all parameters (see
(2.57)) $\\{n,k_{1},\dots,k_{6}\\}$. Using Theorem A.2, it thus suffices to
show $d_{\operatorname{BL}}(\mathfrak{m},\overline{\mathfrak{m}})\to 0$ to
conclude that

$\partial_{2}(\Pi(\mathfrak{m})\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{k_{2}},\Pi(\overline{\mathfrak{m}})\cap\Gamma_{k_{7}}~{}|~{}\mathscr{F}_{k_{2}})\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$

Letting $F$ be the functions in $\mathcal{C}((-2\pi k_{1},0),{\mathbb{C}})$
with max modulus in $[e^{-k_{7}},e^{k_{7}}]$, we can bound

$\displaystyle d_{\operatorname{BL}}(\mathfrak{m},\overline{\mathfrak{m}})$
$\displaystyle\leq(\mathfrak{m}(\Gamma_{k_{7}})+\overline{\mathfrak{m}}(\Gamma_{k_{7}}))$
$\displaystyle\times\sum_{j\in{\mathcal{D}}_{n/k_{1}}}\mathbf{1}\left\\{{\mathscr{R}^{2}_{j}({n}_{1}^{+})}\right\\}d_{\operatorname{BL}}\biggl{(}\int_{0}^{2\pi}\mathfrak{s}_{k_{1},k_{5}}(e^{i\xi-\sqrt{4/\beta}V_{j}^{\prime}}df)\tfrac{d\xi}{2\pi},\mathfrak{s}_{k_{1},k_{5}}(e^{i\alpha_{j}-\sqrt{4/\beta}V_{j}^{\prime}}df)\biggr{)},$

where
$\alpha_{j}\coloneqq\Psi_{n_{1}^{+}}(\theta_{j})-(n_{1}^{+}+1)\theta_{j}$ and
the bounded Lipschitz norm is restricted to $F$. We show in Corollary 10.4
that this distance is bounded by $\mathcal{O}(e^{-(\log
k_{1})^{19/20}})p(V_{j}^{\prime})$ where
$p(V_{j}^{\prime})=\mathbb{P}(W_{j}^{o}\in[-k_{7},\infty)~{}|~{}\mathscr{F}_{n_{1}^{+}}).$
Thus like in (2.57)

$d_{\operatorname{BL}}(\mathfrak{m},\overline{\mathfrak{m}})\xrightarrow[k_{1},n\to\infty]{\mathbb{P}}0.$

For the final step, we show

$\partial_{2}(\iota\\#(\Pi(\mathfrak{n})\cap\Gamma_{k_{7}})~{}|~{}\mathscr{F}_{k_{2}},\iota\\#(\Pi(\overline{\mathfrak{m}})\cap\Gamma_{k_{7}})~{}|~{}\mathscr{F}_{k_{2}})\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$

We again use Theorem A.2 to reduce this to controlling the bounded Lipschitz
norm on $\Gamma_{k_{7}}$. Let $\bar{\iota}\\#\mathfrak{n}$ and
$\bar{\iota}\\#\overline{\mathfrak{m}}$ be the restriction of
$\iota\\#(\mathfrak{n}\cap\Gamma_{k_{7}})$ and
$\iota\\#(\overline{\mathfrak{m}}\cap\Gamma_{k_{7}})$ to
$[0,2\pi]\times\mathcal{C}([-2\pi k_{1},0],{\mathbb{C}})$, and note that the
restriction to $\mathcal{C}([-2\pi k_{1},0],{\mathbb{C}})$ is identical for
both processes except for for a additive random variable, taking value in a
compact ($k_{7}$-dependent) set. From Corollary A.3 and the fact that after
the push forward by $\iota$, the second coordinate in $\Gamma$ is continuously
determined by the third, there is a finite list of non-negative Lipschitz
functions $\\{f_{j}\\}$, depending on $k_{7}$, for which it suffices to show
that

$\bar{\iota}\\#\mathfrak{n}(f_{j})-\bar{\iota}\\#\overline{\mathfrak{m}}(f_{j})\xrightarrow[k_{3},k_{2},k_{1},n\to\infty]{\mathbb{P}}0.$
(2.70)

(We note that the use of $\iota$ was precisely to reduce the collections of
functions $\\{f_{j}\\}$ to a collection that does not depend on $k_{1}$.) Now,
(2.70) follows from Lemma 2.25. ∎

##  3 Interpolation–based regularity arguments

In this section we give a proofs of Proposition 2.6 and Proposition 2.15; both
rely on certain _a priori_ regularity properties of $\varphi_{n},$ but the
second one additionally uses a substantial probabilistic input, Proposition
4.8 in the case $\sigma=1$. The $\sigma=i$ case is substantially simpler; in
effect the regularity is much better for imaginary $\sigma$ owing to the
monotonicity of the Prüfer phases. We introduce the following notation for
working with the case of real $\sigma.$ For any $\theta\in{\mathbb{R}},$
define (with $\delta_{\beta}$ given by half of $\delta$ from Proposition 4.8)

$J(\theta)\coloneqq\left\\{\theta^{\prime}\in\frac{2\pi}{4k_{5}n}{\mathbb{Z}}:|\theta^{\prime}-\theta|\leq\frac{\pi}{nk_{5}^{1-\delta_{\beta}}}\right\\}.$
(3.1)

The deterministic result we need is the following:

######  Lemma 3.1.

For $\sigma=i,$ and any $\theta^{\prime}\leq\theta$

$\varphi_{n}(\theta^{\prime})\leq\varphi_{n}(\theta)+(n+1)(\theta-\theta^{\prime}).$

For $\sigma=1$ if
$\sup_{\theta}\varphi_{n}(\theta)\leq\sqrt{\tfrac{8}{\beta}}m_{n}+k_{6},$ then
there is an absolute constant $C$ so that for all $n,k_{5}$ sufficiently large
with respect to $k_{6},$ and all $\theta\in[0,2\pi]$

$\displaystyle\max_{|\theta^{\prime}-\theta|<\frac{2\pi}{4nk_{5}^{1-\delta_{\beta}}}}\biggl{\\{}e^{\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}}\biggr{\\}}\leq\biggl{(}\frac{2k_{5}}{2k_{5}-1}\biggr{)}\cdot\max_{\theta^{\prime}\in
J(\theta)}\biggl{\\{}e^{\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}}\biggr{\\}}+\frac{Ce^{k_{6}}}{k_{5}^{\delta_{\beta}}}\quad\text{and}$
$\displaystyle\min_{|\theta^{\prime}-\theta|<\frac{2\pi}{4nk_{5}^{1-\delta_{\beta}}}}\biggl{\\{}e^{\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}}\biggr{\\}}\geq\biggl{(}\frac{2k_{5}}{2k_{5}-1}\biggr{)}\cdot\min_{\theta^{\prime}\in
J(\theta)}\biggl{\\{}e^{\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}}\biggr{\\}}-\frac{Ce^{k_{6}}}{k_{5}^{\delta_{\beta}}}.$

###### Proof.

For the first claim, we use that the Prüfer phases
$\theta\mapsto\Psi_{k}(\theta)$ are monotone increasing in $\theta$ (Lemma
2.1. For the second claim, we apply Theorem 8.2 to the polynomial $Q$ such
that $|Q(e^{i\theta})|^{2}=e^{\varphi_{n}(\theta)}$ for all real $\theta$.
This has degree $n$ and the theorem applies with $m=2k_{5}$ and $b\asymp
k_{5}^{1-\delta_{\beta}}.$ ∎

To take advantage of this deterministic result, we then use some first and
second moment estimates which when combined with Lemma 3.1 imply Proposition
2.15. The first of these is essentially a triviality that shows that we can
disregard near maxima that occur near the boundary of an interval
$\widehat{I}_{j},$ more specifially those mesh points that are not contained
in $K=\cup_{j\in\mathcal{D}_{n/k_{1}}}I_{j}$ (recall the definition of $I_{j}$
in (2.23)).

######  Lemma 3.2.

For all $k_{2},k_{4},k_{5},k_{6}$

$\limsup_{n\to\infty}\sum_{j=1}^{4k_{5}n}\mathbf{1}\\{\tfrac{\pi
j}{2k_{5}n}\not\in K\\}\mathbb{E}[\mathbf{1}\\{\mathscr{L}(\tfrac{\pi
j}{2k_{5}n})\\}\mathbf{1}\\{\mathscr{R}(\tfrac{\pi
j}{2k_{5}n})\\}\mathbf{1}\\{\mathscr{G}_{n}\\}~{}|~{}\mathscr{F}_{k_{2}}]\xrightarrow[k_{1}\to\infty]{\mathbb{P}}0.$

###### Proof.

For any $I_{j}$ the fraction of angles $\theta\in I_{j}$ so that
$\theta\not\in K$ vanishes like $\tfrac{\log k_{1}}{k_{1}}$ as
$k_{1}\to\infty.$ In particular, the left-hand-side of the display tends to
$0$ like $O_{k_{p},p\geq 2}((\log k_{1})/k_{1})$ (compare with the proof of
Proposition 2.5). ∎

From here we can also give a quick proof of Proposition 2.6:

###### Proof of Proposition 2.6.

We first observe that on the event $\mathscr{G}_{n},$ globally
$\theta\mapsto\varphi_{n}(\theta)$ is bounded by $k_{6}$, and hence we always
have $W_{j}\leq k_{6}$ for any $j\in\mathcal{D}_{n/k_{1}}.$

Suppose that $\widehat{W}_{j}\geq-k_{7}$, and hence there is a $\theta$ in a
neighborhood of $\widehat{I}_{j}$ at which
$\varphi_{n}(\theta)-\sqrt{\tfrac{8}{\beta}}m_{n}\geq-k_{7}.$ Using the first
conclusion of Lemma 3.1, there must be a $\theta^{\prime}\in J(\theta)$ at
which

$e^{\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}}\geq\biggl{(}\frac{2k_{5}-1}{2k_{5}}\biggr{)}\biggl{(}e^{-k_{7}}-\frac{Ce^{k_{6}}}{k_{5}^{\delta_{\beta}}}\biggr{)}.$

By virtue of Lemma 3.2, we may assume that $\theta^{\prime}\in I_{j}.$ Making
$k_{5}$ large, we conclude

$\varphi_{n}(\theta^{\prime})-\sqrt{\tfrac{8}{\beta}}m_{n}\geq-
k_{7}-o_{k_{5}}(1).$

As $k_{6}$ is much larger that $k_{7},$ this concludes the proof. ∎

The second probabilistic input we need to Proposition 2.15 is that on the
finite sets $J(\theta)$ for $\theta\in I_{j}$ at which $\mathscr{L}(\theta)$
occurs, the process $\varphi$ can be taken nearly constant (or more
specifically its oscillation is no more than $k_{5}^{-\delta_{\beta}}$)
simultaneously for all $\theta$ for which $\mathscr{L}(\theta)$ holds. Define
the event

$\mathscr{J}(\theta)\coloneqq\biggl{\\{}\bigl{(}\max_{\theta^{\prime}\in
J(\theta)}\\{\varphi_{n}(\theta^{\prime})\\}-\min_{\theta^{\prime}\in
J(\theta)}\\{\varphi_{n}(\theta^{\prime})\\}\bigr{)}\leq
k_{5}^{-\delta_{\beta}}\biggr{\\}}.$ (3.2)

Thus we show:

######  Lemma 3.3.

For all $k_{6}$

$\displaystyle\limsup_{n\to\infty}\sum_{j=1}^{4k_{5}n}\mathbb{E}[\mathbf{1}\\{\mathscr{L}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{R}_{j}({n}_{1}^{+})\cap\mathscr{P}_{j}^{\prime}\cap\mathscr{O}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{O}^{\Psi}(\tfrac{\pi
j}{2k_{5}n}))\\}\mathbf{1}\\{\mathscr{J}^{c}(\tfrac{\pi
j}{2k_{5}n})\\}\mathbf{1}\\{\mathscr{G}_{n}\\}~{}|~{}\mathscr{F}_{k_{2}}]$
$\displaystyle\qquad\qquad\qquad\qquad\xrightarrow[k_{5},k_{4},k_{2},k_{1}\to\infty]{\mathbb{P}}0.$

Proposition 2.15 follows immediately from Lemmas 3.3 and 3.1 in the case
$\sigma=1$.

###### Proof.

We may additionally work on the event $\mathscr{G}_{n}^{1},$ using which we
may replace $\varphi_{n}(\theta_{\ell}+\tfrac{\theta}{n})$ by
$\mathfrak{U}_{T_{+}}(\theta)$. The main technical work is contained in
Proposition 4.8. Under this proposition, we have an estimate

$\displaystyle\mathbb{E}[\mathbf{1}\\{\mathscr{L}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{R}_{j}({n}_{1}^{+})\cap\mathscr{P}_{j}^{\prime}\cap\mathscr{O}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{O}^{\Psi}(\tfrac{\pi
j}{2k_{5}n}))\\}\mathbf{1}\\{\mathscr{J}^{c}(\tfrac{\pi
j}{2k_{5}n})\\}\mathbf{1}\\{\mathscr{G}_{n}\\}~{}|~{}\mathscr{F}_{n_{1}^{+}}]$
$\displaystyle\leq
C(k_{6})k_{5}^{-(1+2\delta_{\beta})(1-\delta_{\beta})}\frac{\bigl{(}\sqrt{\tfrac{8}{\beta}}m_{n_{1}^{+}}-\varphi_{n_{1}^{+}}(\tfrac{\pi
j}{2k_{5}n})\bigr{)}\exp\bigl{(}\sqrt{\tfrac{\beta}{2}}\varphi_{n_{1}^{+}}(\tfrac{\pi
j}{2k_{5}n})-2m_{n_{1}^{+}}\bigr{)}}{k_{1}^{+}(\log
k_{1})^{3/2}}\mathbb{E}[\mathscr{R}_{j}({n}_{1}^{+}))\\}~{}|~{}\mathscr{F}_{n_{1}^{+}}].$

We have used here that $\varphi_{n_{1}^{+}}(\tfrac{\pi
j}{2k_{5}n})=\mathfrak{U}_{T_{-}}^{j}(\tfrac{\pi j}{2k_{5}}-n\theta_{j})$, the
initial condtions from (2.35). Bounding above the probability, which proceeds
in the same fashion as (2.18)–(2.22), we have

$\displaystyle\limsup_{k_{1},n\to\infty}$
$\displaystyle\sum_{j=1}^{4k_{5}n}\mathbb{E}[\mathbf{1}\\{\mathscr{L}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{R}_{j}({n}_{1}^{+})\cap\mathscr{P}_{j}^{\prime}\cap\mathscr{O}(\tfrac{\pi
j}{2k_{5}n})\cap\mathscr{O}^{\Psi}(\tfrac{\pi
j}{2k_{5}n}))\\}\mathbf{1}\\{\mathscr{J}^{c}(\tfrac{\pi
j}{2k_{5}n})\\}\mathbf{1}\\{\mathscr{G}_{n}\\}~{}|~{}\mathscr{F}_{k_{2}}]$
$\displaystyle\leq
C(k_{6})k_{5}^{-\delta_{\beta}+2\delta_{\beta}^{2}}\int_{0}^{2\pi}e^{\sqrt{\tfrac{\beta}{2}}(6k_{6}+\varphi_{k_{2}}(\theta))-\log(k_{2})}\bigl{(}C_{\beta}k_{6}+\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{)}d\theta$
$\displaystyle+k_{5}^{-\delta_{\beta}+2\delta_{\beta}^{2}}\int_{0}^{2\pi}e^{\sqrt{\tfrac{\beta}{2}}(\varphi_{k_{2}}(\theta)+k_{6})-\log(k_{2})}\mathbf{1}\left\\{{\varphi_{k_{2}}(\theta)\not\in[s_{k_{2}}^{-},s_{k_{2}}^{+}]}\right\\}\bigl{|}\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta)\bigr{|}d\theta.$

Using Theorem 1.6 we have on sending $k_{2}\to\infty$ that this upper bound
converges almost surely. On afterward sending $k_{5}$ to infinity, the result
follows. ∎

##  4 Bessel bridges

In what follows $\theta_{0}$ will be a fixed angle in $[0,2\pi].$ Recall from
(2.6) that for $\sigma\in\left\\{1,i\right\\},$

$G_{k+1}(\theta_{0})=G_{k}(\theta_{0})-2\Re\\{\sigma\bigl{(}\tfrac{Z_{k}}{\beta_{k}}\bigr{)}e^{i\Psi_{k}(\theta)}\\},\quad
G_{0}(\theta_{0})=\theta_{0}.$

Recall further, see (2.4), (2.6), the complex Brownian motion
$\mathfrak{Z}^{\mathbb{C}}_{t}(\theta_{0})$ with the normalization
$[\mathfrak{Z}_{t}^{\mathbb{C}}(\theta_{0}),\overline{\mathfrak{Z}_{t}^{\mathbb{C}}(\theta_{0})}]=2t,$
and the standard real Brownian motion
$\mathfrak{Z}_{t}=\Re(\sigma\mathfrak{Z}_{t}^{\mathbb{C}}(\theta_{0}))$ for
any $t\geq 0,$ so that
$G_{k}(\theta_{0})=\sqrt{\tfrac{4}{\beta}}\mathfrak{Z}_{H_{k}}(\theta_{0})$
for any $k\geq 0.$

We shall work conditionally on the endpoints $\mathfrak{Z}_{t}$ of various
intervals $[t_{0},t_{1}]$, after which the process is a standard Brownian
bridge. Further conditioning this bridge to lie below the line $t\mapsto\alpha
t,$ on the interval $[t_{0},t_{1}]$ the process $\alpha t-\mathfrak{Z}_{t}$
has the law of a _$3$ –dimensional Bessel process bridge_. It remains a
semimartingale after this conditioning (with respect to the appropriate
augmented filtration), and moreover it is a strong solution to an SDE. We
record these facts and some distributional facts about this Bessel bridge in
the following lemma.

######  Lemma 4.1.

Let $(B_{t}:t\geq 0)$ be a standard real Brownian motion. For any
$\alpha,c_{0},c_{1}>0$ and any $0<t_{0}<t_{1},$ let
$(\mathfrak{X}_{t}:t\in[t_{0},t_{1}])$ be the strong solution of

$\mathfrak{X}_{t}-\mathfrak{X}_{t_{0}}=\alpha(t-t_{0})+(B_{t}-B_{t_{0}})+\int_{t_{0}}^{t}\left(\frac{1}{\mathfrak{X}_{s}-\alpha
s}-\frac{\mathfrak{X}_{s}-(\alpha
s-c_{1})}{t_{1}-s}\right)\,ds,\quad\mathfrak{X}_{t_{0}}=\alpha t_{0}-c_{0}.$

Then $(\mathfrak{Z}_{t}:t\in[t_{0},t_{1}])$ has the same law as
$(\mathfrak{X}_{t}:t\in[t_{0},t_{1}])$ when conditioning on

$\mathfrak{Z}_{t_{0}}=\alpha t_{0}-c_{0},\quad\mathfrak{Z}_{t_{1}}=\alpha
t_{1}-c_{1},\quad{\text{and}}\quad\mathfrak{Z}_{t}\leq\alpha t\quad\text{for
all}\quad t\in[t_{0},t_{1}].$

For any $t\in(t_{0},t_{1}),$ the density of $y_{t}=\alpha t-\mathfrak{X}_{t}$
under the conditioning is given by

$f(u)=Z(t_{1}-t_{0},c_{0},c_{1})\sinh\biggl{(}\frac{c_{0}u}{t-t_{0}}\biggr{)}\sinh\biggl{(}\frac{uc_{1}}{t_{1}-t}\biggr{)}\exp\biggl{(}-\frac{u^{2}}{2(t-t_{0})}-\frac{u^{2}}{2(t_{1}-t)}\biggr{)},\quad
u\geq 0.$

Provided $(c_{0}^{2}+c_{1}^{2})\leq(t_{1}-t_{0})$ we therefore have the
estimates, with $s=\min\\{t-t_{0},t_{1}-t\\}$, for some absolute constant $C$

$\mathbb{P}(\alpha t-\mathfrak{X}_{t}\leq
x)\leq\frac{Cx^{3}}{s^{3/2}}\quad\text{for all}\quad x\leq\sqrt{s}.$

If in addition $\min\\{c_{0},c_{1}\\}\geq 1$ there is a constant so that for
either $i\in\\{0,1\\}$

$\mathbb{P}(\alpha t-\mathfrak{X}_{t}\in du)\leq
Ce^{-(u-c_{i})^{2}/(2|t-t_{i}|)}\frac{udu}{|t-t_{i}|}\quad\text{for all}\quad
u\geq 0.$

###### Proof.

See [RY99, Chapter XI] for a good exposition. Setting $X_{t}=\alpha
t-\mathfrak{Z}_{t},$ the SDE reduces to a standard result on a Brownian motion
conditioned to stay positive (c.f. [RY99, Exercise XI.3.11.2], [Rob13]).

The density is given in [RY99, Chapter XI.3]. The normalizing constant $Z$ is
given by

$Z=\sqrt{\frac{2}{\pi}}\sqrt{\frac{t_{1}-t_{0}}{(t-t_{0})(t_{1}-t)}}\exp\biggl{(}-\frac{c_{0}^{2}}{2(t-t_{0})}+\frac{c_{0}^{2}}{2(t_{1}-t_{0})}-\frac{c_{1}^{2}}{2(t_{1}-t)}+\frac{c_{1}^{2}}{2(t_{1}-t_{0})}\biggr{)}\\!\operatorname{csch}\biggl{(}\frac{c_{0}c_{1}}{t_{1}-t_{0}}\biggr{)}.$

Under the assumption on $c_{0}^{2}+c_{1}^{2}$ we may bound this normalizing
constant above as

$Z\leq C\biggl{(}\frac{t_{1}-t_{0}}{c_{0}c_{1}\sqrt{s}}\biggr{)}.$

Using $\sinh(x)\leq xe$ for $x\leq 1,$ we arrive at a bound on the density $f$
for $u\leq\sqrt{s},$

$f(u)\leq\frac{3Cu^{2}}{s^{3/2}}\quad\text{for all}\quad u\leq\sqrt{s}.$

For the upper tail, we assume without loss of generality that $t-t_{0}\leq
t_{1}-t.$ In this case we also assume that $\min\\{c_{0},c_{1}\\}\geq 1.$ If
$u\leq(t_{1}-t_{0})/(\max_{i}c_{i})$ then using $\sinh(x)\leq e^{x}/2$ for one
of the terms and $\sinh(x)\leq ex$ for the other, for some $C>0$

$f(u)\leq\frac{Cu}{s}e^{-(u-c_{0})^{2}/(2s)}du.$

If on the other hand $u\leq(t_{1}-t_{0})/(\max_{i}c_{i})$ then $Z\leq
Cu/\sqrt{s}$ and so we conclude again

$f(u)\leq\frac{Cu}{s}e^{-(u-c_{0})^{2}/(2s)}du.$

∎

We also note that a Bessel bridge has good oscillation properties on short
time windows, provided its endpoints are not brought close to the barrier:

######  Lemma 4.2.

Suppose that $(\mathfrak{X}_{t}:t\in[0,1])$ has the law of a $3$-dimensional
Bessel bridge with $\mathfrak{X}_{0}=x_{0}$ and $\mathfrak{X}_{1}=x_{1}$. Then
if $x_{1},x_{0}\geq 1$, there is an absolute constant $c>0$ so that

$\mathbb{P}\biggl{(}\sup_{t\in[0,1]}|\mathfrak{X}_{t}-(x_{1}-x_{0})t-x_{0}|\geq
t+c\biggr{)}\leq\exp(-t^{2}/c).$

###### Proof.

We can realize the Bessel bridge by taking a Brownian Bridge
$\mathfrak{Z}_{t}$ with the same endpoints and conditioning it to remain
positive. Let $\mathcal{H}$ be the event that this Brownian bridge is
positive. It then suffices to prove that

$\mathbb{P}\biggl{(}\biggl{\\{}\sup_{t\in[0,1]}|\mathfrak{Z}_{t}-(x_{1}-x_{0})t-x_{0}|\geq
t+c\biggr{\\}}~{}|~{}\mathcal{H}\biggr{)}\leq\exp(-t^{2}/c).$

This is a standard fact for $\mathfrak{Z}_{t}$ without the conditioning, i.e.
there is a $c>0$ sufficiently large that

$\mathbb{P}\biggl{(}\biggl{\\{}\sup_{t\in[0,1]}|\mathfrak{Z}_{t}-(x_{1}-x_{0})t-x_{0}|\geq
t+c\biggr{\\}}\biggr{)}\leq\exp(-t^{2}/c).$

The event $\mathcal{H}$ has probability bounded below by some absolute
constant (indeed it is possible to compute it), but note that by monotonicity
it is bounded below by the same probability where we take $x_{1}=x_{0}=1,$
which in any case is some number $p>0$, and so

$\mathbb{P}\biggl{(}\biggl{\\{}\sup_{t\in[0,1]}|\mathfrak{Z}_{t}-(x_{1}-x_{0})t-x_{0}|\geq
t+c\biggr{\\}}~{}|~{}\mathcal{H}\biggr{)}\leq\exp(-t^{2}/c)/p.$

Then increasing $c$, the claimed bound follows. ∎

As our first application, we show that conditionally on the process lying
slightly below a concave barrier, the process in fact tends to stay in the
entropic envelope. Recall the event $\mathscr{B}_{n,k_{2},k_{6}}$ from (2.9).

######  Lemma 4.3.

On the event that $\mathfrak{Z}_{H_{k_{2}}}\in\sqrt{2}(\log k_{2}-[(\log
k_{2})^{0.49},(\log k_{2})^{0.51}])$ and that
$\mathfrak{Z}_{H_{n}}\in\sqrt{2}m_{n}+[-k_{6},k_{6}],$ there is a constant
$c>0$ sufficiently small so that for all $n\gg k_{2}\gg k_{4}\gg k_{5}$
sufficiently large

$\mathbb{P}(\widehat{\mathscr{R}}(\theta)^{c}\cap\mathscr{B}_{n,k_{2},k_{6}}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})\leq\frac{(\sqrt{2}\log
k_{2}-\mathfrak{Z}_{H_{k_{2}}})e^{-c(\log k_{5})^{2}}}{\log n}.$

###### Proof.

We will work conditionally on $\mathfrak{Z}_{H_{k_{2}}}$ and
$\mathfrak{Z}_{H_{n}}$ throughout the proof, and we will work on the
$\sigma(\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$–measurable event in
the statement of the lemma.

We recall that on the event $\mathscr{B}_{n,k_{2},k_{6}},$ we have (see (2.8))

$\mathfrak{Z}_{H_{2^{k}}}\leq\sqrt{\beta}k_{6}+\sqrt{2}A_{2^{k}}^{\ll}\quad\text{for
all }\log_{2}k_{2}\leq k\leq\log_{2}n.$ (4.1)

From [CMN18, Lemma A.5], the probability of this event is bounded above by

$\mathbb{P}(\eqref{eq:aha-1}|\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})\leq
C_{\beta}\frac{(\sqrt{2}\log
k_{2}-\mathfrak{Z}_{H_{k_{2}}})_{+}k_{6}}{\log(n/k_{2})}.$ (4.2)

We can fill in this barrier and slightly increase its height and define the
event

$\mathfrak{Z}_{t}\leq\sqrt{2}(1-\tfrac{3}{4}\tfrac{\log\log n}{\log
n})t+g(t),\quad g(t)\coloneqq\biggl{(}\frac{t(H_{n}-t+(\log
k_{5})^{50})}{H_{n}}\biggr{)}^{1/50}\quad\text{for all }H_{k_{2}}\leq t\leq
H_{n}.$ (4.3)

By a simple union bound estimate over each interval $[H_{2^{k-1}},H_{2^{k}}]$
for all $k_{5}$ sufficiently large

$\mathbb{P}(\eqref{eq:aha0}^{c}|\eqref{eq:aha-1},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})\leq
e^{-c(k_{6})(\log k_{5})^{2}}.$ (4.4)

Note that we therefore have
$\mathbb{P}(\eqref{eq:aha0}^{c}\cap\eqref{eq:aha-1}|\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$
is the same order as the bound claimed in the lemma, and it suffices to, going
forward, bound
$\mathbb{P}(\widehat{\mathscr{R}}(\theta)^{c}\cap\eqref{eq:aha0}|\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$
(which will be lower order for $k_{4}\gg k_{5}$).

Recall that under $\mathbb{P},$ conditional on
$\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}},$ the process

$\mathfrak{B}_{t}=\mathfrak{Z}_{t}-\mathfrak{Z}_{H_{k_{2}}}-\frac{\mathfrak{Z}_{H_{n}}-\mathfrak{Z}_{H_{k_{2}}}}{H_{n}-H_{k_{2}}}(t-H_{k_{2}})$

is a bridge starting and ending an $0.$ Define a change of measure by

$\displaystyle\frac{d{\mathbb{Q}}}{d\mathbb{P}}$
$\displaystyle\coloneqq\exp\left(\int_{H_{k_{2}}}^{H_{n}}g^{\prime}(t)d\mathfrak{Z}_{t}-\frac{1}{2}\int_{H_{k_{2}}}^{H_{n}}(g^{\prime}(t))^{2}dt\right)$
(4.5)
$\displaystyle=\exp\left(F(\mathfrak{Z}_{H_{n}},\mathfrak{Z}_{H_{k_{2}}})-\int_{H_{k_{2}}}^{H_{n}}\bigl{\\{}g^{\prime\prime}(t)\mathfrak{B}_{t}+\tfrac{(g^{\prime}(t))^{2}}{2}\bigr{\\}}dt\right),$

for some (linear) functional $F.$ Then, without conditioning on its endpoints,
under ${\mathbb{Q}},$ the process $\mathfrak{Z}_{t}-g(t)$ is a standard
Brownian motion for $t\in[H_{k_{2}},H_{n}],$ and after conditioning this
process is a Brownian bridge. Moreover, we see that the conditional
Radon–Nikodym derivative is just given by (4.5) with $F=0.$

Estimating the probability
${\mathbb{Q}}(\widehat{\mathscr{R}}^{c}(\theta)~{}|~{}\eqref{eq:aha0})$ is
straightforward, given that there is an explicit probabilistic description for
the conditional process. So, our goal is to show that
$\mathbb{P}(\widehat{\mathscr{R}}^{c}(\theta)\cap\eqref{eq:aha0})$ is not much
larger.

$\displaystyle\mathbb{P}(\widehat{\mathscr{R}}^{c}(\theta)\cap\eqref{eq:aha0}~{}|~{}|\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$
$\displaystyle={\mathbb{Q}}\bigl{[}\tfrac{d\mathbb{P}}{d{\mathbb{Q}}}\mathbf{1}\left\\{{\widehat{\mathscr{R}}^{c}(\theta)\cap\eqref{eq:aha0}}\right\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\bigr{]}$
(4.6)
$\displaystyle={\mathbb{Q}}\bigl{[}\tfrac{d\mathbb{P}}{d{\mathbb{Q}}}\mathbf{1}\left\\{{\widehat{\mathscr{R}}^{c}(\theta)}\right\\}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\bigr{]}{\mathbb{Q}}\bigl{[}\eqref{eq:aha0}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\bigr{]}.$

Conditioning on (4.3), $\mathfrak{Z}_{H_{k_{2}}},$ and $\mathfrak{Z}_{H_{n}},$
$\mathfrak{Z}_{t}-g(t)$ has the law of a Bessel bridge under ${\mathbb{Q}}$.
Thus under ${\mathbb{Q}}$ we can use the estimates in Lemma 4.1, to estimate
the probability of $\widehat{\mathscr{R}}(\theta)$ under ${\mathbb{Q}}$.

We first establish that under the conditional measure
${\mathbb{Q}}(\cdot~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$,
the process $\mathfrak{Z}_{t}$ is not outside the entropic barrier of half the
size at _integer times_ up to $H_{n}-k_{4}+1$, (c.f. (2.12))

$\tilde{A}_{t}^{n,\pm}\coloneqq t+\begin{cases}-0.5t^{1/2\mp 2/5}&\text{ if
}t\leq\frac{1}{2}\log(n),\\\ -0.5(\log n-t)^{1/2\mp 2/5}-\frac{3}{4}\log\log
n&\text{ if }\frac{1}{2}\log(n)<t\leq\log(n).\end{cases}$ (4.7)

For bounding the event of exceeding the entropic barrier we use the
summability of the powers in 4.1, to conclude

$\displaystyle{\mathbb{Q}}(\mathfrak{Z}_{t}\notin[\tilde{A}_{t}^{n,-},\tilde{A}_{t}^{n,+}]\text{
for some integer $t\leq
H_{n}-k_{4}+1$}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$
$\displaystyle\leq C\bigl{(}(\log k_{2})^{-1/5}+k_{4}^{-1/5}\bigr{)}.$

We may then fill in the gaps by again using union bound together with the
oscillation estimate Lemma 4.2 over each interval to conclude for some larger
constant $C>0$

${\mathbb{Q}}(\widehat{\mathscr{R}}(\theta)|\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})\geq
e^{-C(\log k_{2})^{-1/5}-Ck_{4}^{-1/5}}$ (4.8)

for all $k_{2},k_{4}$ sufficiently large. We will show that we can reduce to
this case by bounding the expression
${\mathbb{Q}}\bigl{[}\bigl{(}\tfrac{d\mathbb{P}}{d{\mathbb{Q}}}\bigr{)}^{2}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\bigr{]}.$
Let $\alpha=\sqrt{2}(1-\tfrac{3}{4}\tfrac{\log\log n}{\log
n})-\frac{\mathfrak{Z}_{H_{n}}-\mathfrak{Z}_{H_{k_{2}}}}{H_{n}-H_{k_{2}}}=O(\tfrac{(\log
k_{2})^{0.51}+k_{6}}{\log n}),$ and which is positive for all $k_{2}$ and $n$
sufficiently large. Under ${\mathbb{Q}}$ and conditioned on
$\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}$, the process

$\displaystyle\mathfrak{X}_{t}$
$\displaystyle=-\mathfrak{B}_{t}+g(t)+\alpha(t-H_{k_{2}})+\sqrt{2}(1-\tfrac{3}{4}\tfrac{\log\log
n}{\log n})H_{k_{2}}-\mathfrak{Z}_{H_{k_{2}}}$
$\displaystyle=-\mathfrak{Z}_{t}+g(t)+\sqrt{2}(1-\tfrac{3}{4}\tfrac{\log\log
n}{\log n})t$

is a Brownian bridge and the event (4.3) becomes

$\mathfrak{X}_{t}\geq 0\quad\text{for all }\quad t\in[H_{k_{2}},H_{n}].$

Thus, conditionally on (4.3), $(\mathfrak{X}_{t}:t\in[H_{k_{2}},H_{n}])$ has
the law of a Bessel bridge (c.f. Lemma 4.1).

The conditional Radon–Nikodym derivative can be expressed in terms of
$\mathfrak{X}$ by

$\frac{d\mathbb{P}}{d{\mathbb{Q}}}=\exp\left(\int_{H_{k_{2}}}^{H_{n}}\bigl{\\{}-g^{\prime\prime}(t)\mathfrak{X}_{t}+g^{\prime\prime}(t)(\alpha(t-H_{k_{2}})+\sqrt{2}(1-\tfrac{3}{4}\tfrac{\log\log
n}{\log
n})H_{k_{2}}-\mathfrak{Z}_{H_{k_{2}}})-\tfrac{(g^{\prime}(t))^{2}}{2}\bigr{\\}}dt\right).$
(4.9)

We can estimate the function $g$ for a sufficiently large absolute constant
$L>0,$

$|g^{\prime\prime}(t)|\leq L(t^{-99/50}\wedge(H_{n}-t+\log k_{5})^{-99/50}),$
(4.10)

and we also observe that $g$ is concave, recall (4.3). From the concavity of
$g$ and the convexity of the exponential, we can therefore bound

$\biggl{(}\frac{d\mathbb{P}}{d{\mathbb{Q}}}\biggr{)}^{2}\leq\exp\left(\int_{H_{k_{2}}}^{H_{n}}-2g^{\prime\prime}(t)\mathfrak{X}_{t}dt\right).$
(4.11)

From Lemma 4.1, we have a subgaussian estimate on $\mathfrak{X}_{t}$. Recall
the subgaussian norm

$\|X\|_{\psi_{2}}=\inf\\{t\geq
0:\mathbb{E}[e^{X^{2}/t^{2}}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}]\leq
2\\},$

such that

$\|\mathfrak{X}_{t}\|_{\psi_{2}}\leq\begin{cases}(\log
k_{2})^{0.51}+\sqrt{t}&t\leq\tfrac{1}{2}H_{n}\\\ k_{6}+\sqrt{H_{n}-t}&t\leq
H_{n}-k_{4}\\\ \end{cases}$

Hence using Jensen’s inequality,

$\biggl{\|}\int_{H_{k_{2}}}^{H_{n}}-2g^{\prime\prime}(t)\mathfrak{X}_{t}dt\biggr{\|}_{\psi_{2}}\leq\int_{H_{k_{2}}}^{H_{n}}2g^{\prime\prime}(t)\|\mathfrak{X}_{t}\|_{\psi_{2}}dt.$

Splitting the integral and using the boung on $g^{\prime\prime}$ and
$\|\mathfrak{X}_{t}\|_{\psi_{2}}$, we have

$\int_{H_{k_{2}}}^{H_{n}}2g^{\prime\prime}(t)\|\mathfrak{X}_{t}\|_{\psi_{2}}dt\leq
C(k_{6},L)\biggl{(}(\log k_{2})^{-47/100}+k_{4}^{-48/100}\biggr{)}.$

In summary to control of the Radon–Nikodym derivative we conclude for all
$k_{4}$ and $k_{2}$ sufficiently large there is an absolute constant $C>0$ so
that

${\mathbb{Q}}\biggl{(}\biggl{(}\frac{d\mathbb{P}}{d{\mathbb{Q}}}\biggr{)}^{2}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\biggr{)}\leq
e^{Ck_{4}^{-24/25}+C(\log k_{2})^{-47/50}}.$ (4.12)

Hence if we turn to (4.6), then using Cauchy-Schwarz, we get that for all
$k_{2}$ and $k_{4}$ sufficiently large

${\mathbb{Q}}\bigl{[}\tfrac{d\mathbb{P}}{d{\mathbb{Q}}}\mathbf{1}\left\\{{\widehat{\mathscr{R}}^{c}(\theta)}\right\\}~{}|~{}\eqref{eq:aha0},\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}}\bigr{]}\leq(\log
k_{2})^{-1/5}+k_{4}^{-1/5}.$ (4.13)

Finally, the ${\mathbb{Q}}$ probability of (4.3), conditionally on
$\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}},$ is the probability that the
Brownian bridge $\mathfrak{X}_{t}$ stays positive for all time. This
probability is in fact explicit [KS91, (3.40)]:

$\displaystyle{\mathbb{Q}}(\eqref{eq:aha0}|\mathfrak{Z}_{H_{k_{2}}},\mathfrak{Z}_{H_{n}})$
$\displaystyle=1-\exp\left(-\frac{2\mathfrak{X}_{H_{k_{2}}}\mathfrak{X}_{H_{n}}}{H_{n}-H_{k_{2}}}\right)$
(4.14) $\displaystyle\leq C\frac{(\sqrt{2}\log
k_{2}-\mathfrak{Z}_{H_{k_{2}}})(\log k_{5})}{\log(n)},$

for all $n,k_{2}$ sufficiently large and some $C>0$. For $n$ large, this
contribution is negligible in comparison to (4.4). Hence by (4.6) and (4.13),
the proof is complete. ∎

By a similar argument, we can actually estimate the density of the Bessel
process endpoint killed when it exits an entropic envelope.

######  Lemma 4.4.

For notational convenience, let $s=H_{n_{1}^{+}}$. Fix $p\geq 2$. On the event
that $\mathfrak{Z}_{H_{k_{2}}}\in\sqrt{2}(\log k_{2}-[(\log
k_{2})^{0.49},(\log k_{2})^{0.51}])$ and $y\in[A_{s}^{p-1,-},A_{s}^{p-1,+}]$

$\begin{aligned}
&\mathbb{P}(\mathscr{R}^{p}_{j}(n_{1}^{+})\cap\\{\sqrt{2}m_{n_{1}^{+}}-\mathfrak{Z}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})\\\
&=\sqrt{\frac{2}{\pi}}\frac{y}{n_{1}^{+}}\bigl{(}\sqrt{2}H_{k_{2}}-\mathfrak{Z}_{H_{k_{2}}}\bigr{)}\exp\bigl{(}\sqrt{2}y+\sqrt{2}\mathfrak{Z}_{H_{k_{2}}}-\log
k_{2}+o_{k_{2}}\bigr{)}\end{aligned}.$

Furthermore for all $y\in[A_{s}^{p,-},A_{s}^{p,+}]$, the density is upper
bounded by the right hand side. With $y\in[A_{s}^{p-1,-},A_{s}^{p-1,+}]$ if we
introduce $\mathscr{V}_{j}^{\prime}$ (see (2.61)) as the alteration of (2.26),
where we only put the entropic envelope restriction at $t$ in the set
$\\{H_{2^{k}}:k\in{\mathbb{N}}\\}$, we furthermore have

$\displaystyle\mathbb{P}(\bigl{(}\mathscr{R}^{2}_{j}(n_{1}^{+})\bigr{)}^{c}\cap\mathscr{V}_{j}^{\prime}\cap\\{\sqrt{2}m_{n_{1}^{+}}-\mathfrak{Z}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})$
$\displaystyle\leq
o_{k_{2}}\times\sqrt{\frac{2}{\pi}}\frac{y}{n_{1}^{+}}\bigl{(}\sqrt{2}H_{k_{2}}-\mathfrak{Z}_{H_{k_{2}}}\bigr{)}\exp\bigl{(}\sqrt{2}y+\sqrt{2}\mathfrak{Z}_{H_{k_{2}}}-\log
k_{2}\bigr{)}.$

###### Proof.

Define the event $\mathscr{E}$ that

$\mathfrak{X}_{t}\coloneqq\sqrt{2}\bigl{(}1-\tfrac{3}{4}\tfrac{\log
s}{s}\bigr{)}t-\mathfrak{Z}_{t}\geq 0\quad\text{for all}\quad
t\in[H_{k_{2}},s],$ (4.15)

and note that $\mathscr{R}^{p}_{j}(n_{1}^{+})\subset\mathscr{E}$.

Define the change of measure

$\frac{d{\mathbb{Q}}}{d\mathbb{P}}=\exp\bigl{(}\sqrt{2}\bigl{(}1-\tfrac{3}{4}\tfrac{\log
s}{s}\bigr{)}(\mathfrak{Z}_{s}-\mathfrak{Z}_{H_{k_{2}}})-\bigl{(}1-\tfrac{3}{4}\tfrac{\log
s}{s}\bigr{)}^{2}(s-H_{k_{2}})\bigr{)}.$

Under ${\mathbb{Q}}$, $\mathfrak{X}_{t}$ is a standard Brownian motion and
conditionally on $\mathscr{E}$, $\mathfrak{X}_{t}$ is a Bessel-3 process on
$[H_{k_{2}},s]$. As this determines the Radon–Nikodym derivative, we have

$\displaystyle\mathbb{P}(\mathscr{R}^{p}_{j}(n_{1}^{+})\cap\\{\sqrt{2}m_{n_{1}^{+}}-\mathfrak{Z}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})$
(4.16)
$\displaystyle={\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})\cap\\{\mathfrak{X}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})$
$\displaystyle\times\exp\bigl{(}-s+\tfrac{3}{2}\log
s+\sqrt{2}y+\sqrt{2}\mathfrak{Z}_{H_{k_{2}}}-H_{k_{2}}+o_{n}\bigr{)}.$

Define for convenience $t=s-H_{k_{2}}$ and
$x=\sqrt{2}\bigl{(}1-\tfrac{3}{4}\tfrac{\log
s}{s}\bigr{)}H_{k_{2}}-\mathfrak{Z}_{H_{k_{2}}}.$ Under
${\mathbb{Q}}((\cdot)~{}|~{}\mathfrak{Z}_{H_{k_{2}}})$, $\mathfrak{X}_{t}$ has
a gaussian density of mean $\mathfrak{X}_{H_{k_{2}}}$ and variance $t$.
However by the restriction on $y$, the gaussian trivializes, and so

${\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})\cap\\{\mathfrak{X}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})={\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y)\sqrt{\frac{1}{2\pi
t}}(1+o_{n}).$

To produce the upper bound on the density, we use the inclusion
$\mathscr{R}^{p}_{j}(n_{1}^{+})\subset\mathscr{E}$ and then bound, using
(4.14),

${\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y)\leq{\mathbb{Q}}(\mathscr{E}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y)=1-\exp(-2xy/t).$

Hence we conclude, from combining everything with (4.16),

$\displaystyle\mathbb{P}(\mathscr{R}^{p}_{j}(n_{1}^{+})\cap\\{\sqrt{2}m_{n_{1}^{+}}-\mathfrak{Z}_{s}\in[y,y+dy]\\}~{}|~{}\mathfrak{Z}_{H_{k_{2}}})$
$\displaystyle\leq\sqrt{\frac{2}{\pi}}\frac{yx}{t^{3/2}}\exp\bigl{(}-s+\tfrac{3}{2}\log
s+\sqrt{2}y-(x^{2}+y^{2})/2t+\sqrt{2}\mathfrak{Z}_{H_{k_{2}}}-H_{k_{2}}+o_{n}\bigr{)}$
$\displaystyle\leq\sqrt{\frac{2}{\pi}}\frac{yx}{n_{1}^{+}}\exp\bigl{(}\sqrt{2}y+\sqrt{2}\mathfrak{Z}_{H_{k_{2}}}-\log
k_{2}+o_{k_{2}}\bigr{)}.$

For the lower bound, we condition on $\mathscr{E}$ and express

${\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y)={\mathbb{Q}}(\mathscr{E}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y){\mathbb{Q}}(\mathscr{R}^{p}_{j}(n_{1}^{+})~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y,\mathcal{E}).$

Thus it suffices to give a vanishing upper bound on

${\mathbb{Q}}\bigl{(}\bigl{(}\mathscr{R}^{p}_{j}(n_{1}^{+})\bigr{)}^{c}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y,\mathcal{E}\bigr{)}=o_{k_{2}}.$

Under the extra assumption that $y$ is far from the entropic window (as
assumed in the statement of the lemma), this is now similar to what is shown
in Lemma 4.3, so we briefly summarize the argument. We start by bounding the
probability that the Bessel bridge escapes the entropic envelope at times in
$H_{2^{k}}$ with $k\in{\mathbb{N}}$, and then between two integer times use a
gaussian tail bound for the oscillation Lemma 4.2.

The same oscillation lemma shows that with $p=2$

${\mathbb{Q}}\bigl{(}\mathscr{E}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y,\mathscr{V}_{j}^{\prime}\bigr{)}=1-o_{k_{2}}$

which implies the second claim since

$\displaystyle{\mathbb{Q}}\bigl{(}\bigl{(}\mathscr{R}^{2}_{j}(n_{1}^{+})\bigr{)}^{c}\cap\mathscr{V}_{j}^{\prime}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y\bigr{)}$
$\displaystyle=(1+o_{k_{2}}){\mathbb{Q}}\bigl{(}\bigl{(}\mathscr{R}^{2}_{j}(n_{1}^{+})\bigr{)}^{c}\cap\mathscr{V}_{j}^{\prime}\cap\mathscr{E}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y\bigr{)}$

and therefore

${\mathbb{Q}}\bigl{(}\bigl{(}\mathscr{R}^{2}_{j}(n_{1}^{+})\bigr{)}^{c}\cap\mathscr{V}_{j}^{\prime}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y,\mathscr{E}\bigr{)}\leq{\mathbb{Q}}\bigl{(}\bigl{(}\mathscr{R}^{2}_{j}(n_{1}^{+})\bigr{)}^{c}~{}|~{}\mathfrak{Z}_{H_{k_{2}}},\mathfrak{X}_{s}=y,\mathscr{E}\bigr{)}.$

∎

###  4.1 The coarse oscillation bound

In this section, we consider a single $j\in\mathcal{D}_{n/k_{1}}$, and hence
we write simply the barrier event

$\displaystyle{\mathscr{R}}$ $\displaystyle=\left\\{\forall~{}\log k_{2}\leq
t\leq\log\widehat{n}_{1}:\sqrt{\tfrac{8}{\beta}}A_{t}^{4,-}\leq\sqrt{\tfrac{4}{\beta}}\mathfrak{Z}_{t}(\theta_{0})\leq\sqrt{\tfrac{8}{\beta}}A_{t}^{4,+}\right\\}.$
(4.17)

In this section, our goal is to estimate the probability of the oscillation
event

$\displaystyle\widehat{\mathscr{O}}=|\Psi_{n_{1}^{+}}(\theta^{-})-\Psi_{n_{1}^{+}}(\theta^{+})|\leq\tfrac{k_{1}(\log
k_{1})^{50}}{k_{1}^{+}}$ (4.18)

with $\theta^{\pm}=\theta_{0}+\frac{\pm 2\pi k_{1}}{n},$ when we condition on
$\mathscr{R}.$ Let $(\mathscr{H}_{t}:t\geq 0)$ be the join of the filtrations
(recalling (2.2))

$\bigl{(}(\mathfrak{Z}^{\mathbb{C}}_{s}:0\leq s\leq t):t\geq
0\bigr{)}\quad\text{and}\quad\bigl{(}({X}_{k},{Y_{k}},\Gamma_{k}:\forall~{}k~{},H_{k}\leq
t):t\geq 0\bigr{)},$

where we further augment by $\mathfrak{Z}_{\log{\widehat{n}_{1}}}.$ Note that
$G_{k}$ and $\Psi_{k}$ are adapted to $\mathscr{H}_{H_{k}}$ for any
$k\in{\mathbb{N}}.$ Let $\tau$ be any $\mathscr{H}$ stopping time such that
for all $\log k_{2}\leq t\leq\tau,$ $\mathfrak{Z}_{t}$ satisfies

$\sqrt{\tfrac{4}{{\beta}}}\mathfrak{Z}_{t}\in\sqrt{\tfrac{8}{{\beta}}}[{A}_{t}^{4,-},{A}_{t}^{4,+}].$
(4.19)

Then (see (4.17)), if $\tau$ is just the first time (4.19) fails, then on
$\mathscr{R}$ it follows $\tau>\log\widehat{n}_{1}.$

We will let $\mathcal{B}$ be the event that $\mathfrak{Z}_{t}\leq\sqrt{2}t$
for all $t\geq H_{k_{2}}$ and
$\mathfrak{Z}_{\log{\widehat{n}_{1}}}\in[\mathcal{A}_{\log\widehat{n_{1}}}^{4,-},\mathcal{A}_{\log\widehat{n}_{1}}^{4,+}].$
Then conditionally on $\mathcal{B},$ we can use Lemma 4.1 to compute the
behavior of increments of $\mathfrak{Z}.$

######  Lemma 4.5.

Set $\Delta_{k+1}\coloneqq\mathfrak{Z}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}_{\tau\wedge H_{k}}.$ For $k_{2}\leq k\leq\log
n_{1}^{+},$ there is a constant $C>0$ so that for all $k_{2}$ and $k_{1}$
sufficiently large,

$\displaystyle\mathbb{E}[\Delta_{k+1}|\mathscr{H}_{H_{k}},\mathcal{B}]=(\sqrt{2}\pm
c(k_{1},k_{2}))\mathbb{E}[(\tau\wedge H_{k+1}-\tau\wedge
H_{k})|\mathscr{H}_{H_{k}},\mathcal{B}]$
$\displaystyle\mathbb{E}[\Delta_{k+1}^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]\leq\mathbb{E}[(\tau\wedge
H_{k+1}-\tau\wedge
H_{k})|\mathscr{H}_{H_{k}},\mathcal{B}]+C(\mathbb{E}[(\tau\wedge
H_{k+1}-\tau\wedge H_{k})^{2}|\mathscr{H}_{H_{k}},\mathcal{B}],$

where the meaning of the first line is LHS bounded above and below by the RHS,
choosing signs appropriately, and where

$c(k_{1},k_{2})\leq C((\log k_{2})^{-1/18}+(\log k_{1})^{-1/100}).$

for some constant $C>0.$

###### Proof.

We can express the increment $\Delta_{k+1},$ using Lemma 4.1, as

$\displaystyle\Delta_{k+1}$ $\displaystyle=\sqrt{2}(\tau\wedge
H_{k+1}-\tau\wedge H_{k})+B_{\tau\wedge H_{k+1}}-B_{\tau\wedge H_{k}}$ (4.20)
$\displaystyle+\int_{\tau\wedge H_{k}}^{\tau\wedge
H_{k+1}}\left(\frac{1}{\mathfrak{Z}_{s}-\sqrt{2}s}-\frac{\mathfrak{Z}_{s}-\mathfrak{Z}_{\log\widehat{n}_{1}}-\sqrt{2}(s-\log\widehat{n}_{1})}{\log\widehat{n}_{1}-s}\right)\,ds\eqqcolon\Delta_{k+1}^{\prime}$

where $B_{(\cdot)}$ is a $\mathscr{H}$–adapted standard Brownian motion. The
increment $\Delta_{k+1}^{\prime}$ we can control using the barriers. By
construction, for $s\leq\tau$ for all $k_{1}$ and $n$ sufficiently large

$-\sqrt{2}(\log k_{2})^{17/18}\leq\mathfrak{Z}_{s}-\sqrt{2}s\leq-\sqrt{2}(\log
k_{2})^{1/18}.$

For $\tau\geq s\geq\frac{1}{2}\log n,$ we can also bound

$-\sqrt{2}(\log
n-s)^{17/18}\leq\mathfrak{Z}_{s}-\sqrt{2}(s-\tfrac{3}{4}\log\log
n)\leq-\sqrt{2}(\log n-s)^{1/18}.$

In particular, for $k\leq\log n_{1}^{+}$ and all $n$ sufficiently large

$|\Delta_{k+1}^{\prime}|\leq(\tau\wedge H_{k+1}-\tau\wedge
H_{k})\biggl{(}(\log k_{2})^{-1/18}+2(\log k_{1})^{-1/100}\biggr{)}.$ (4.21)

As $B_{\tau\wedge(\cdot)}$ is a martingale,

$\mathbb{E}[\Delta_{k+1}|\mathscr{H}_{H_{k}},\mathcal{B}]=\sqrt{\tfrac{8}{\beta}}\mathbb{E}[(\tau\wedge
H_{k+1}-\tau\wedge
H_{k})|\mathscr{H}_{H_{k}},\mathcal{B}]+\mathbb{E}[\Delta_{k+1}^{\prime}|\mathscr{H}_{H_{k}},\mathcal{B}],$

and using the bound (4.21), the claim follows. For the second moment, we use

$\sqrt{\mathbb{E}[\Delta_{k+1}^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]}\leq\sqrt{\mathbb{E}[(\Delta_{k+1}-\Delta_{k+1}^{\prime})^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]}+\sqrt{\mathbb{E}[(\Delta_{k+1}^{\prime})^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]}.$
(4.22)

For the first term, we have the exact formula

$\mathbb{E}[(\Delta_{k+1}-\Delta_{k+1}^{\prime})^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]=2\mathbb{E}[(\tau\wedge
H_{k+1}-\tau\wedge
H_{k})^{2}|\mathscr{H}_{H_{k}},\mathcal{B}]+\mathbb{E}[\tau\wedge
H_{k+1}-\tau\wedge H_{k}|\mathscr{H}_{H_{k}},\mathcal{B}].$

Hence using (4.21) and (4.22), the bound for the second moment follows. ∎

######  Proposition 4.6.

There is a $C_{\beta}$ so that for all $k_{1}$ sufficiently large, depending
on $k_{2},$ and all $n$ sufficiently large, depending on $k_{1}$, so that
uniformly in $\theta_{0}$

$\mathbb{P}[\widehat{\mathscr{O}}^{c}\cap\mathscr{R}\cap\mathscr{G}_{n}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\leq
C_{\beta}(\log k_{1})^{-50}\quad\operatorname{a.s.}$

As the conditional probability of $\mathcal{B}$ is explicit, we conclude:

######  Corollary 4.7.

There is a constant $C_{\beta}$ so that for all $n,k_{1}$ sufficiently large
(random)

$\mathbb{P}[\widehat{\mathscr{O}}^{c}\cap\mathscr{R}\cap\mathscr{G}_{n}~{}|~{}\mathscr{H}_{H_{k_{2}}}]\leq
C_{\beta}\frac{\left(\sqrt{2}\log
k_{2}-\sqrt{\tfrac{\beta}{4}}\varphi_{k_{2}}(\theta_{0})+k_{6}\right)_{+}\left(\sqrt{\tfrac{8}{\beta}}m_{n}-G_{\widehat{n}_{1}}(\theta_{0})\right)_{+}}{(\log
k_{1})^{50}\log(\widehat{n}_{1}/k_{2})}.$

###### Proof.

We multiply the result of Proposition 4.6 by
$\mathbb{P}[\mathcal{B}~{}|~{}\mathscr{H}_{H_{k_{2}}}],$ which is the
probability that a Brownian bridge stays below a straight barrier (see
(4.14)). ∎

###### Proof of Proposition 4.6.

Let
$\psi_{k}(\theta-\theta_{0})\coloneqq\Psi_{k}(\theta)-\Psi_{k}(\theta_{0})$
for any $k\in{\mathbb{N}}$ for any $\theta\geq\theta_{0}.$ We will show that
for all $k_{1}$ and $n$ sufficiently large (depending on $k_{2}$) there is a
constant $C_{\beta}$ so that

$\mathbb{P}[\\{\psi_{n_{1}^{+}}(\theta_{+}-\theta_{0})>\tfrac{k_{1}(\log
k_{1})^{50}}{4k_{1}^{+}}\\}\cap\mathscr{R}\cap\mathscr{G}_{n}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\leq
C_{\beta}(\log k_{1})^{-50}\quad\operatorname{a.s.}$ (4.23)

The analogous bound for
$|\psi_{n_{1}^{+}}(\theta_{-}-\theta_{0})|=-\psi_{n_{1}^{+}}(\theta_{-}-\theta_{0})$
holds by the same argument after making appropriate sign changes. We just show
(4.23). The proof will use calculus computations contained in Section 7.

We will just write $\theta$ for $\theta_{+}-\theta_{0}.$ Let $\tau$ be the
first time $t\geq\log k_{2}$ that either (4.19) fails, or for some $H_{k}\leq
t$ where $k_{2}\leq k$ either $\psi_{H_{k}}(\theta)\geq\pi$ or

$\max_{t\in[H_{k},H_{k+1}]}|\mathfrak{Z}_{t}^{\mathbb{C}}-\mathfrak{Z}_{H_{k}}^{\mathbb{C}}|^{2}>\frac{32\log
k}{k+1}\quad\text{or}\quad|\sqrt{\Gamma_{H_{k}}}-\beta_{H_{k}}|>4\sqrt{\log
H_{k}}.$ (4.24)

Define

$Z_{k}^{\tau}\coloneqq-\sqrt{\frac{k+1}{2}}\bigl{(}\mathfrak{Z}_{\tau\wedge
H_{k+1}}^{\mathbb{C}}-\mathfrak{Z}_{\tau\wedge
H_{k}}^{\mathbb{C}}\bigr{)},\quad\text{ and
$\gamma_{k}^{\tau}=\frac{Z_{k}^{\tau}}{\sqrt{|Z_{k}^{\tau}|^{2}+\Gamma_{k}}}.$}$

For any $\theta>0,$ let $\psi_{k}^{\tau}(\theta)$ solve (c.f. (1.9))

$\psi_{k+1}^{\tau}(\theta)=\psi_{k}^{\tau}(\theta)+\begin{cases}\theta-2\Im\left(\log(1-\gamma_{k}^{\tau}e^{i\psi_{k}^{\tau}(\theta)})-\log(1-\gamma_{k}^{\tau})\right)&\text{if
}H_{k}<\tau,\\\ 0&\text{else},\end{cases}$

and $\psi_{0}^{\tau}(\theta)=\theta.$ Then on the event
$\mathscr{R}\cap\mathscr{G}_{n}$ (compare (2.7)) if $\\{\max_{k_{2}\leq k\leq
n_{1}+}\psi^{\tau}_{k}(\theta)<\pi\\}$ we must actually have $\tau>\log
n_{1}^{+}.$ Thus for any $t>0$

$\mathbb{P}[\\{\psi_{n_{1}^{+}}(\theta)>t\\}\cap\mathscr{R}\cap\mathscr{G}_{n}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\leq\mathbb{P}[\\{\max_{k_{2}\leq
k\leq
n_{1}+}\psi^{\tau}_{k}(\theta)>t\\}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\quad\operatorname{a.s.}$
(4.25)

On the event $\tau>\log n_{1}^{+},$
$\psi_{n_{1}^{+}}^{\tau}(\theta)=\psi_{n_{1}^{+}}(\theta).$ By construction,
$\psi^{\tau}_{k}(\theta)\geq 0$ almost surely (see Lemma 2.1).

From Corollary 7.2 we have that for $k_{2}$ sufficiently large, the relative
Prüfer phases (recall (1.9)) satisfy for some absolute constant $C$ and any
$H_{k}<\tau$

$\displaystyle\psi_{k+1}^{\tau}(\theta)-\psi_{k}^{\tau}(\theta)$
$\displaystyle\leq\theta+2\Im\biggl{(}\frac{(e^{i\psi_{k}^{\tau}(\theta)}-1)Z_{k}^{\tau}}{\sqrt{\Gamma_{k}}}\biggr{)}+\frac{C\psi_{k}^{\tau}(\theta)|Z_{k}^{\tau}|^{2}}{\Gamma_{k}}.$
(4.26)

_The case $\sigma=1$:_ In this case (4.26) becomes for $H_{k}<\tau$

$\displaystyle\psi_{k+1}^{\tau}(\theta)-\psi_{k}^{\tau}(\theta)\leq\theta-$
$\displaystyle\frac{2}{\sqrt{\Gamma_{k}}}\frac{\sqrt{k+1}}{\sqrt{2}}\biggl{(}\sin(\psi_{k}^{\tau}(\theta))(\mathfrak{Z}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}_{\tau\wedge H_{k}})\biggr{)}$ (4.27) $\displaystyle-$
$\displaystyle\frac{2}{\sqrt{\Gamma_{k}}}\frac{\sqrt{k+1}}{\sqrt{2}}\biggl{(}(\cos(\psi_{k}^{\tau}(\theta))-1)\Im(\sigma(\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge H_{k}}))\biggr{)}$
$\displaystyle+$
$\displaystyle\frac{C\psi_{k}^{\tau}(\theta)|\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge H_{k}}|^{2}}{\Gamma_{k}}.$

Then computing the conditional expectation of this increment given
$\mathscr{H}_{H_{k}}$ using Lemma 4.5, and using that
$\psi_{k}^{\tau}(\theta)\in[0,\pi)$ for some $C_{\beta}>0$,

$\displaystyle\mathbb{E}[\psi_{k+1}^{\tau}(\theta)-\psi_{k}^{\tau}(\theta)~{}|~{}\mathscr{H}_{H_{k}},\mathcal{B}]\leq\theta+\frac{C_{\beta}\psi_{k}^{\tau}(\theta)}{(k+1)^{2}}\quad\text{for}\quad
k_{2}\leq k\leq n_{1}^{+}.$ (4.28)

The remainder of the real case will be covered by the argument for the
imaginary cases, but the argument for the real case is simpler and given
below. If we define the increasing function $k\mapsto P_{k}$ by the recurrence

$P_{k+1}\coloneqq P_{k}+\theta+\frac{C_{\beta}P_{k}}{(k+1)^{2}},\quad\text{for
all $k\geq 0$ and}\quad P_{0}\coloneqq\theta,$ (4.29)

then $\psi_{k}^{\tau}(\theta)-P_{k}$ is a supermartingale started at $0.$
Hence for all $t\geq 0,$

$\displaystyle\mathbb{P}[\sup_{k_{2}\leq k\leq
n_{1}^{+}}\psi_{k}^{\tau}(\theta)>t+P_{n_{1}^{+}}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]$
$\displaystyle\leq\mathbb{P}[\sup_{k_{2}\leq k\leq
n_{1}^{+}}(\psi_{k}^{\tau}(\theta)-P_{k})>t~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]$
(4.30) $\displaystyle\leq\frac{\psi_{k_{2}}^{\tau}(\theta)+P_{n_{1}^{+}}}{t},$

with the final inequality following from the same argument as Doob’s
inequality. The recurrence for $P_{k}$ (4.29) is easily solved, and it can be
checked that

$P_{k}\leq C_{\beta}(k+1)\theta\qquad\text{for any}\quad k\leq n_{1}^{+}.$
(4.31)

for some other sufficiently large $C_{\beta}.$ Hence as $\theta\leq
Cn_{1}^{-1},$ we have $P_{n_{1}^{+}}\leq C_{\beta}e^{-(\log
k_{1})^{(29/30)}}.$ Thus (4.30) and (4.25) imply (4.23).

_The imaginary case $\sigma=i$:_ In this case (4.26) becomes for $H_{k}<\tau$

$\displaystyle\psi_{k+1}^{\tau}(\theta)-\psi_{k}^{\tau}(\theta)\leq\theta+$
$\displaystyle\frac{2}{\sqrt{\Gamma_{k}}}\frac{\sqrt{k+1}}{\sqrt{2}}\biggl{(}(1-\cos(\psi_{k}^{\tau}(\theta)))(\mathfrak{Z}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}_{\tau\wedge H_{k}})\biggr{)}$ (4.32) $\displaystyle\mp$
$\displaystyle\frac{2}{\sqrt{\Gamma_{k}}}\frac{\sqrt{k+1}}{\sqrt{2}}\biggl{(}\sin(\psi_{k}^{\tau}(\theta))\Im(\sigma(\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge H_{k}}))\biggr{)}$
$\displaystyle+$
$\displaystyle\frac{C\psi_{k}^{\tau}(\theta)|\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge
H_{k+1}}-\mathfrak{Z}^{{\mathbb{C}}}_{\tau\wedge H_{k}}|^{2}}{\Gamma_{k}}.$

The $\mp$ depends on the sign of $\sigma.$ We note that there will also be a
sign change in the $(1-\cos)$ term. For either of $\sigma=i$ it will be
necessary to consider the (less advantageous) case considered on account of
needing to consider the case $\theta_{-}$ (in which $\theta<0$).

Appling Lemma 4.5 and bounding the cosine,

$\displaystyle\mathbb{E}[\psi_{k+1}^{\tau}(\theta)-\psi_{k}^{\tau}(\theta)~{}|~{}\mathscr{H}_{H_{k}},\mathcal{B}]\leq\theta+\frac{C_{\beta}(\psi_{k}^{\tau}(\theta))^{2}}{(k+1)}+\frac{C_{\beta}\psi_{k}^{\tau}(\theta)}{(k+1)^{2}}\quad\text{for}\quad
k_{2}\leq k\leq n_{1}^{+}.$ (4.33)

Let $\eta(x)=(\log\tfrac{1}{x})^{-100}e^{(\log\tfrac{1}{x})^{29/30}}$ for
$x\in(0,1)$ and define a stopping time $\vartheta$ as the first $k\geq k_{2}$
such that

$\psi_{k}^{\tau}(\theta)>\eta(\tfrac{k+1}{n_{1}})(k+1)\theta.$

Then the stopped process $\psi_{k\wedge\vartheta}^{\tau}$ satisfies for
$k_{2}\leq k\leq n_{1}^{+}$,

$\mathbb{E}[\psi_{\vartheta\wedge(k+1)}^{\tau}(\theta)-\psi_{\vartheta\wedge
k}^{\tau}(\theta)~{}|~{}\mathscr{H}_{H_{k}},\mathcal{B}]\leq\theta+C_{\beta}\eta^{2}(\tfrac{k+1}{n_{1}})(k+1)\theta^{2}+\frac{C_{\beta}\psi_{\vartheta\wedge
k}^{\tau}(\theta)}{(k+1)^{2}}.$ (4.34)

If we define the increasing function $k\mapsto P_{k}$ by the recurrence

$P_{k+1}\coloneqq
P_{k}+\theta+C_{\beta}\eta^{2}(\tfrac{k+1}{n_{1}})(k+1)\theta^{2}+\frac{C_{\beta}P_{k}}{(k+1)^{2}},\quad\text{for
all $k\geq 0$ and}\quad P_{0}\coloneqq\theta,$ (4.35)

then $\psi_{k\wedge\vartheta}^{\tau}(\theta)-P_{k\wedge\vartheta}$ is a
supermartingale started at $0.$ The recurrence for $P_{k}$ is easily solved
explicitly, and in particular there is a constant $C_{\beta}$ sufficiently
large that for any $0\leq k\leq n_{1}^{+},$

$k\theta\leq P_{k}-P_{0}\leq
C_{\beta}\sum_{\ell=0}^{k-1}\bigl{\\{}\theta+\eta^{2}(\tfrac{\ell+1}{n_{1}})(\ell+1)\theta^{2}\bigr{\\}}$
(4.36)

for all $k_{1}$ sufficiently large. If we take $c$ as the maximum of
$x\eta^{2}(x)$ on $[0,1],$ then we can futher bound

$P_{k}-P_{0}\leq(1+2c)C_{\beta}k\theta.$ (4.37)

Moreover, for all $t\geq 0$ and any $m\leq n_{1}^{+},$ by the same argument as
in Doob’s inequality

$\displaystyle\mathbb{P}[\sup_{k_{2}\leq k\leq
m}\psi_{k\wedge\vartheta}^{\tau}(\theta)>t+P_{m}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]$
$\displaystyle\leq\mathbb{P}[\sup_{k_{2}\leq k\leq
m}(\psi_{k\wedge\vartheta}^{\tau}(\theta)-P_{k\wedge\vartheta})>t~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]$
(4.38)
$\displaystyle\leq\frac{(\psi_{k_{2}}^{\tau}(\theta)-P_{k_{2}})+P_{m}}{t}.$

We can use this bound to control the probability that
$\vartheta\in[2^{\ell-1},2^{\ell}].$ For this to happen for some
$\ell\geq\log_{2}k_{2}$ we must have for some $c_{\beta}$ sufficiently small

$\sup_{k_{2}\leq k\leq
2^{\ell}}\psi_{k\wedge\vartheta}^{\tau}(\theta)>P_{2^{\ell}}(1+c_{\beta}\eta(\tfrac{2^{\ell-1}}{n_{1}})).$

Hence summing over $\log_{2}k_{2}\leq\ell\leq\log_{2}n_{1}^{+},$ and using
(4.37) and (4.38) and increasing $C_{\beta}$ as needed between the
inequalities,

$\mathbb{P}(\vartheta\leq n_{1}^{+})\leq
C_{\beta}\sum_{\ell=\log_{2}k_{2}}^{\log_{2}(n_{1}^{+})}\frac{\psi_{k_{2}}^{\tau}(\theta)+2^{\ell}\theta}{2^{\ell}\theta\eta(\tfrac{2^{\ell-1}}{n_{1}})}\leq
C_{\beta}\biggl{(}\frac{\psi_{k_{2}}^{\tau}(\theta)}{k_{2}\theta\eta(\tfrac{k_{2}}{n_{1}})}+\frac{(\log\tfrac{n_{1}^{+}}{n_{1}})^{\tfrac{1}{30}}}{\eta(\tfrac{n_{1}^{+}}{n_{1}})}\biggr{)}.$

On the event $\\{\vartheta>n_{1}^{+}\\}\cap\mathscr{G}_{n},$ we thus have that
$\tau>n_{1}^{+}$ and

$\psi_{n_{1}^{+}}(\theta)=\psi_{n_{1}^{+}}^{\tau}(\theta)\leq\eta(\tfrac{n_{1}^{+}+1}{n_{1}})(n_{1}^{+}+1)\theta.$

Hence if we apply (4.38) again with $t=(\log k_{1})^{50}P_{n_{1}^{+}},$ we
conclude that for all $k_{1}$ sufficiently large

$\mathbb{P}[\sup_{k_{2}\leq k\leq n_{1}^{+}}\psi_{k}^{\tau}(\theta)>(\log
k_{1})^{50}P_{n_{1}^{+}}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\leq
C_{\beta}\biggl{(}\frac{\psi_{k_{2}}^{\tau}(\theta)}{k_{2}\theta\eta(\tfrac{k_{2}}{n_{1}})}+\frac{(\log\tfrac{n_{1}^{+}}{n_{1}})^{\tfrac{1}{30}}}{\eta(\tfrac{n_{1}^{+}}{n_{1}})}\biggr{)}+2\frac{\psi_{k_{2}}^{\tau}(\theta)+P_{n_{1}^{+}}}{(\log
k_{1})^{50}P_{n_{1}^{+}}}.$

To conclude the proof, we observe that
$\theta\mapsto\psi_{k_{2}}^{\tau}(\theta)$ is a continuously differentiable
function, and

$|\psi_{k_{2}}^{\tau}(\theta)|\leq\bigl{\\{}\sup_{\theta_{0}\in[0,2\pi]}|(\psi_{k_{2}}^{\tau})^{\prime}(\theta_{0})|\bigr{\\}}\theta.$

Hence on taking $n\to\infty$ the $\psi_{k_{2}}^{\tau}$ terms tend to $0$
almost surely, and we conclude

$\limsup_{n\to\infty}\mathbb{P}[\sup_{k_{2}\leq k\leq
n_{1}^{+}}\psi_{k}^{\tau}(\theta)>(\log
k_{1})^{50}P_{n_{1}^{+}}~{}|~{}\mathscr{H}_{H_{k_{2}}},\mathcal{B}]\leq
C_{\beta}\biggl{(}\frac{(\log\tfrac{n_{1}^{+}}{n_{1}})^{\tfrac{1}{30}}}{\eta(\tfrac{n_{1}^{+}}{n_{1}})}\biggr{)}+2(\log
k_{1})^{-50},$

which completes the proof by how $\eta$ was chosen. ∎

###  4.2 The fine oscillation bound

In this section, we develop an estimate of continuity for the real part of the
field, where we consider a high value of the field
$\mathfrak{U}^{j}_{T_{+}}(\theta^{\prime})$ and then give a continuity
estimate for $\mathfrak{U}^{j}_{T_{+}}(\theta)$ for
$\theta\in[\theta^{\prime},\theta^{\prime}+\theta_{0}]$ for small
$\theta_{0}.$ Without loss of generality we will take $\theta^{\prime}=0.$

######  Proposition 4.8.

($\sigma=1$). We suppose that for $\alpha>4,\theta_{0}>0$ are given and
satisfy

$(\log k_{1})^{-\alpha/2}\leq\theta_{0}\leq\varepsilon$ (4.39)

where the constant $\varepsilon$ is a small positive constant depending on
$\beta>0$ to be determined. We will condition on
$(\mathfrak{L}_{T_{-}}(\theta):\theta)$. We assume that the oscillation of the
initial conditions are small in that

$\max_{|\theta|\leq\theta_{0}}|\mathfrak{L}^{j}_{T_{-}}(\theta)-\mathfrak{L}^{j}_{T_{-}}(0)|\leq(\log
k_{1})^{-\alpha}.$ (4.40)

We consider large endpoints in the sense that

$\sqrt{\tfrac{8}{\beta}}m_{n}-k_{6}\leq\mathfrak{U}^{j}_{T_{+}}(0)\leq\sqrt{\tfrac{8}{\beta}}m_{n}+k_{6}.$
(4.41)

Then there are $\delta_{j}=\delta_{j}(\beta)>0$ for $j\in\\{1,2\\}$ and a
constant $C_{\beta}$ sufficiently large so that for any fixed set $S$ in
$[0,\theta_{0}]$ of cardinality at most $e^{\theta_{0}^{-\delta_{1}}}$, the
event

$\mathscr{O}_{*}\coloneqq\\{\sup_{\theta\in
S}|\mathfrak{U}^{j}_{T_{+}}(\theta)-\mathfrak{U}^{j}_{T_{+}}(0)|\leq\theta_{0}^{\delta_{1}}\quad\text{and}\quad\Im\bigl{(}\mathfrak{L}^{j}_{T_{+}}(\theta_{0})-\mathfrak{L}^{j}_{T_{+}}(0)\bigr{)}\leq\theta_{0}^{\delta_{1}}\\}$

satisfies on the event (4.40)

$\displaystyle\mathbb{P}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)\cap\eqref{eq7:lb}$
$\displaystyle~{}|~{}(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}$
$\displaystyle\leq C_{\beta}(k_{6})\frac{\theta_{0}^{1+\delta_{2}}(\log
k_{5})\bigl{(}\sqrt{\tfrac{8}{\beta}}m_{n_{1}^{+}}-\mathfrak{U}_{T_{-}}^{j}(0)\bigr{)}\exp\bigl{(}\sqrt{\tfrac{\beta}{2}}\mathfrak{U}_{T_{-}}^{j}(0)-2m_{n_{1}^{+}}\bigr{)}}{k_{1}^{+}(\log
k_{1})^{3/2}}.$

######  Remark 4.9.

An extension of the argument shows that in the case that $\alpha=\infty$
(which is to say the initial conditions are $0$) we may in fact bound the
$\delta_{1}$–Hölder exponent on the interval by a constant with the same
probability (up to constants). This is by applying a chaining argument, and
effectively applying this proposition repeatedly to control the process on
intervals of size $\theta_{0}{2^{-k}}$ for $k=0,1,2,\dots$. Step $1$ would
remain the same (albeit with $\mathscr{O}_{*}$ being the Hölder continuity
event). Steps 2-5 (control on the difference of the imaginary part) should be
generalized from the difference of the imaginary part over the interval
$[0,\theta_{0}]$ to the interval $[\theta_{1},\theta_{2}]$, but essentially no
details change. Step $6$ would change similarly. Finally, the chaining would
be done.

###### Proof.

We let
$d\mathfrak{X}_{t}=\sqrt{\tfrac{4}{\beta}}\Re(e^{i\Im\mathfrak{L}_{t}(0)}d\mathfrak{W}_{t}^{j})$
and let
$d\mathfrak{B}_{t}=\sqrt{\tfrac{4}{\beta}}\Im(e^{i\Im\mathfrak{L}_{t}(0)}d\mathfrak{W}_{t}^{j}),$
for $t\in[T_{-},T_{+}]$. We also set the initial conditions
$\mathfrak{B}_{T_{-}}=0$ and $\mathfrak{X}_{T_{-}}=\mathfrak{L}_{T_{-}}(0)$.

We let $\mathscr{H}$ be the filtration generated by $\mathfrak{X}$ and
$\mathfrak{B}$ with $\mathfrak{X}_{T_{+}}$ and
$(\mathfrak{L}^{j}_{T_{-}}{(\theta)}:\theta)$ adjoined. For a large integer
absolute constant $k_{*}$ we set $T_{*}=T_{+}-k_{*}$ and we condition on the
event that we remain below the concave barrier

$\mathfrak{U}^{j}_{t}(0)\leq\sqrt{\tfrac{8}{\beta}}\bigl{(}m_{n}+t-T_{+}+2\bigl{(}{T_{+}-t+(\log
k_{5})^{50}}\bigr{)}^{1/50}\bigr{)}\bigr{)}\eqqcolon\ell(t)\quad\text{for all
}t\in[T_{-},T_{*}].$ (4.42)

This event is above the concave barrier in $\mathscr{P}_{j}^{\prime}(0).$ We
shall condition on the event in (4.42)

As in the definitions and manipulations between (4.5)–(4.11) from Lemma 4.3,
we define a measure $\mathbb{Q}$ which is mutually absolutely continuous with
respect to $\mathbb{P}$ and which flattens the curvature in (4.42). For this
measure, there is a $(\mathbb{Q},\mathscr{H})$–Brownian motion
$(X_{t}:t\in[T_{-},T_{+}])$ such that

$d\mathfrak{X}_{t}=dX_{t}+\biggl{(}-\ell^{\prime}(t)+\frac{\mathbf{1}\left\\{{t\leq
T_{*}}\right\\}}{\ell(t)+\mathfrak{X}_{t}}+\frac{\mathfrak{X}_{T_{+}}-\mathfrak{X}_{t}+\ell(T_{*})-\ell(t)}{T_{+}-t}\biggr{)}dt,\quad\mathfrak{X}_{T_{-}}=-\mathfrak{U}^{j}_{T_{-}}(0).$
(4.43)

In the above, we have extended the definition of $\ell$ for $t\geq T_{*}$ by
taking it constant and equal to $\ell(T_{*})$. Under $\mathbb{Q}$,
$\mathfrak{B}$ remains a $(\mathbb{Q},\mathscr{H})$–Brownian motion. Moreover
from (4.11), we have a bound on the Radon–Nikodym derivative.

$\frac{d\mathbb{P}}{d\mathbb{Q}}\leq\exp\biggl{(}\int_{T_{-}}^{T_{*}}\\!\\!\\!\\!\\!\\!-\ell^{\prime\prime}(t)(\ell(t)+\mathfrak{X}_{t})dt\biggr{)}$
(4.44)

We also introduce $\mathfrak{D}_{t}=\ell(t\wedge T_{*})+\mathfrak{X}_{t}$,
which under $\mathbb{Q}$ is an $\mathscr{H}$–adapted Bessel bridge with
endpoints

$\mathfrak{D}_{T_{-}}\leq
2\sqrt{\tfrac{8}{\beta}}\exp\bigl{(}\tfrac{9}{10}(\log
k_{1})^{29/30}\bigr{)}\quad\text{and}\quad\mathfrak{D}_{T_{+}}\leq\sqrt{\tfrac{8}{\beta}}\bigl{(}((\log
k_{5})^{50}+k_{*})^{1/50}+k_{6}\bigr{)}.$

##### Step 1: reduction to a conditional probability.

We will show in this section that the problem can be reduced to showing

${\mathbb{Q}}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray}\bigr{)}\leq\theta_{0}^{1+\delta}$
(4.45)

for some $\delta,\delta_{1}>0$ ($\delta_{1}$ hides in the definition of
$\mathscr{O}_{*}$) depending on $\beta$ and all $k_{4},k_{5}$ bigger than some
constant depending only $\beta$. Suppose we have established (4.45). Using the
change of measure, we have

$\displaystyle\mathbb{P}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray}\bigr{)}$
$\displaystyle={\mathbb{Q}}\left[\frac{d\mathbb{P}}{d\mathbb{Q}}\mathbf{1}\left\\{{\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)}\right\\}~{}\middle|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray}\right].$

Applying Hölder’s inequality, there is $\lambda>1$ sufficiently large that

$\mathbb{P}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray}\bigr{)}\leq{\mathbb{Q}}\left(\exp\biggl{(}\int_{T_{-}}^{T_{*}}\\!\\!\\!\\!\\!\\!-\lambda\ell^{\prime\prime}(t)\mathfrak{D}_{t}dt\biggr{)}\right)^{1/\lambda}\theta_{0}^{1+\delta/2}.$

Controlling the Radon–Nikodym derivative is the same argument as the argument
between (4.11) and (4.12). We have for some absolute constant $L>0$

$-\ell^{\prime\prime}(t)\leq L(T_{+}-t+(\log k_{5})^{50})^{-99/50}$

and from Lemma 4.1 for some constant $C_{\beta}$

$\|\mathfrak{D}_{t}\|_{\psi_{2}}\leq C_{\beta}((\log k_{5})+\sqrt{T_{+}-t}),$

where $\|\cdot\|_{\psi_{2}}$ is the subgaussian norm with respect to the
conditional probability measure given as
$\mathbb{Q}(\cdot~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray})$.
Hence by convexity of the norm

$\biggl{\|}\int_{T_{-}}^{T_{*}}\\!\\!\\!\\!\\!\\!-\ell^{\prime\prime}(t)\mathfrak{D}_{t}dt\biggr{\|}_{\psi_{2}}\leq\int_{T_{-}}^{T_{*}}\\!\\!\\!\\!\\!\\!\bigl{\|}-\ell^{\prime\prime}(t)\mathfrak{D}_{t}dt\bigr{\|}_{\psi_{2}}\leq
C_{\beta}(\log k_{5})^{-24}.$

Thus we have after increasing $C_{\beta}$ for all $\lambda\geq 1$

${\mathbb{Q}}\left(\exp\biggl{(}\int_{T_{-}}^{T_{*}}\\!\\!\\!\\!\\!\\!-\lambda\ell^{\prime\prime}(t)\mathfrak{D}_{t}dt\biggr{)}\right)^{1/\lambda}\leq\exp\bigl{(}\lambda
C_{\beta}(\log k_{5})^{-24}\bigr{)},$

which is bounded by $2$ for all $k_{5}$ small (depending on $\lambda$, which
in turn depends only on $\beta$).

Thus we have shown

$\mathbb{P}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta),\eqref{eq7:ray}\bigr{)}\leq\theta_{0}^{1+\delta}$
(4.46)

for some constant $\delta>0$ depending only on $\beta$ and all $k_{4},k_{5}$
bigger than some constant depending only on $\beta$. The event (4.42) contains
$\mathscr{P}_{j}^{\prime}(0)$, and so

$\mathbb{P}\bigl{(}\mathscr{O}_{*}^{c}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}\leq\theta_{0}^{1+\delta}\mathbb{P}\bigl{(}\eqref{eq7:ray}~{}|~{}\mathfrak{U}^{j}_{T_{+}}(0),(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}.$

We then take expectation on both sides of the equation over
$\mathfrak{U}^{j}_{T_{+}}(0)$ satisfying (4.41). Let $\mathbb{M}$ be the
change of measure that flattens the linear part of $\ell$ (with respect to the
conditional measure
$\mathbb{P}(\cdot~{}|~{}(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta))$)

$\frac{d\mathbb{M}}{d\mathbb{P}}=\exp\biggl{(}\sqrt{\tfrac{\beta}{2}}(\mathfrak{U}_{T_{+}}^{j}(0)-\mathfrak{U}_{T_{-}}^{j}(0))-(T_{+}-T_{-})\biggr{)},$

under which

$t\mapsto\mathfrak{U}_{t}^{j}(0)-\mathfrak{U}_{T_{-}}^{j}(0)-\sqrt{\tfrac{8}{\beta}}(t-T_{-})$

is a speed–$(\tfrac{4}{\beta})$ Brownian motion. Note that on (4.41) the
change of measure $\frac{d\mathbb{M}}{d\mathbb{P}}$ is controlled up to
constants by

$\frac{d\mathbb{M}}{d\mathbb{P}}\geq
C(k_{6})\exp\biggl{(}\sqrt{\tfrac{\beta}{2}}(\sqrt{\tfrac{8}{\beta}}m_{n}-\mathfrak{U}_{T_{-}}^{j}(0))-(T_{+}-T_{-})\biggr{)}$

In particular, we have

$\displaystyle\mathbb{P}\bigl{(}\eqref{eq7:ray}\cap\eqref{eq7:lb}~{}|~{}(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}$
$\displaystyle\leq\frac{C(k_{6})}{k_{1}^{+}}\exp\biggl{(}\sqrt{\tfrac{\beta}{2}}\mathfrak{U}_{T_{-}}^{j}(0)-2m_{n_{1}^{+}}\biggr{)}\mathbb{M}\bigl{(}\eqref{eq7:ray}\cap\eqref{eq7:lb}~{}|~{}(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}.$

Using barrier estimates as in [CMN18, Corollary A.6] we can bound the
$\mathbb{M}$–probability for values of $\mathfrak{U}^{j}_{T_{-}}(0)$ given by
$\mathscr{P}_{j}^{\prime}(0)$

$\mathbb{M}\bigl{(}\eqref{eq7:ray}\cap\eqref{eq7:lb}~{}|~{}(\mathfrak{L}^{j}_{T_{-}}(\theta):\theta)\bigr{)}\leq\frac{C(k_{6})(\log
k_{5})\bigl{(}\sqrt{\tfrac{8}{\beta}}m_{n_{1}^{+}}-\mathfrak{U}_{T_{-}}^{j}(0)\bigr{)}}{(\log
k_{1})^{3/2}}.$

This concludes the reduction of the lemma to (4.45).

##### Step 2: Finding a good event on which the slope of the Bessel bridge is
tame.

We introduce a random variable to control the slopes of the Bessel bridge:

$\mathfrak{u}\coloneqq\max_{k\geq k_{*}:T_{+}-k\geq
T_{-}}k^{0.49}\times\biggl{|}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{T_{+}-k}}{k}\biggr{|}.$

Conditioning on $\mathfrak{D}_{T_{*}}$, by the Gaussian tail bound for the
Brownian bridge increment $\mathfrak{D}_{T_{+}}-\mathfrak{D}_{T_{*}}$ and the
Gaussian tail bound for the Bessel bridge increment (c.f. Lemma 4.1)
$\mathfrak{D}_{T_{*}}-\mathfrak{D}_{T_{+}-k}$, there is a Gaussian tail bound
for $\mathfrak{u}$ of the form

$\mathbb{Q}\left(\biggl{|}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{T_{+}-k}}{k}\biggr{|}>C\sqrt{\tfrac{4}{\beta}}(1+x)~{}\middle|~{}\mathscr{H}_{T_{-}}\right)\leq
Ce^{-kx^{2}},$

for some $C$ sufficiently large and all $k\geq k_{*}.$ Thus, summing in $k$ we
may assume that the event

$E_{0}\coloneqq\\{\mathfrak{u}\leq\tfrac{\widehat{\delta}}{3}\bigl{(}\log(1/\theta_{0})\bigr{)}^{0.51}\\}$
(4.47)

occurs with $(\mathbb{Q}~{}|~{}\mathscr{H}_{T_{-}})$–conditional probability
$1-C_{\beta}e^{-c_{\beta}\widehat{\delta}^{2}(\log(1/\theta_{0}))^{1.01}}$ for
all $\widehat{\delta},\theta_{0}>0$ sufficiently small.

##### Step 3: Finding a good event with small oscillations.

The oscillations of $\mathfrak{X}_{t}$ must be controlled, especially near the
endpoint $t=T_{+},$ to control its contribution to the diffusions
$d\mathfrak{X}_{t}+id\mathfrak{B}_{t}.$ These oscillations are ultimately
controlled by those of the underlying Brownian motion ${X}_{t}$. Define the
event $E_{1}$, for $\widehat{\delta}$ to be determined as a function of
$\beta$,

$\max\biggl{\\{}\frac{|X_{u}-X_{s}|}{|u-s|^{1/4}},\frac{|\mathfrak{X}_{u}-\mathfrak{X}_{s}|}{|u-s|^{1/4}}\biggr{\\}}\leq\tfrac{\widehat{\delta}}{3}\bigl{(}\log(1/\theta_{0})^{0.51}(T_{+}-u+1)^{0.49}\bigr{)},\forall~{}T_{+}\geq
u\geq s\geq T_{-},|u-s|\leq 1.$

We may bound this probability by reducing the statement to a union bound over
sets

$\operatorname{CP}_{k}\coloneqq\left\\{(u,s):|T_{+}-u-k|\leq 2,|T_{+}-s-k|\leq
2\right\\}$

for integer $k.$

First we will need a basic input: from oscillation theory of Brownian motion,
there is an absolute constant $C>0$ so that for all $x>0,$

$\mathbb{Q}\biggl{(}\sup_{(u,s)\in\operatorname{CP}_{k}}\frac{|X_{u}-X_{s}|}{|u-s|^{1/4}}>\sqrt{\tfrac{4}{\beta}}C(1+x)~{}\bigg{|}~{}\mathscr{H}_{T_{-}}\biggr{)}\leq
e^{-x^{2}}.$ (4.48)

Taking a union bound over $k$, we conclude that with
$(\mathbb{Q}~{}|~{}\mathscr{H}_{T_{-}})$–conditional probability
$1-C_{\beta}e^{-c_{\beta}\widehat{\delta}^{2}(\log(1/\theta_{0}))^{2}}$ for
all $\widehat{\delta},\theta_{0}>0$ sufficiently small,

$\frac{|X_{u}-X_{s}|}{|u-s|^{1/4}}\leq\tfrac{\widehat{\delta}}{3}\bigl{(}\log(1/\theta_{0})^{0.51}(T_{+}-u+1)^{0.49}\bigr{)},\quad\forall~{}T_{+}\geq
u\geq s\geq T_{-},|u-s|\leq 1.$ (4.49)

To control the oscillations of $\mathfrak{X},$ it suffices to control the
oscillations of $\mathfrak{D}$ in its place, as for any $T_{+}\geq u\geq s\geq
T_{-}$ with $|u-s|\leq 1$

$|\mathfrak{X}_{u}-\mathfrak{X}_{s}|\leq|\mathfrak{D}_{u}-\mathfrak{D}_{s}|+\bigl{(}\max_{T_{-}\leq
t\leq T_{+}}|\ell^{\prime}(t)|\bigr{)}|u-s|,$

and $\ell^{\prime}$ remains bounded.

For the control on $\mathfrak{D}$ we consider separately the cases of $k\leq
k_{*}-2$ and $k\geq k_{*}-2$. In the former case, if we condition on the value
of $\mathfrak{X}_{T_{*}}$ then the process
$\bigl{(}\mathfrak{X}_{t}:t\in[T_{*},T_{+}]\bigr{)}$ is a
speed–$({\tfrac{4}{\beta}})$ Brownian bridge. Its slope can be controlled by
$\mathfrak{u}$, and after removing its slope, the same bound (4.48) holds,
with possibly different constants, and so taking a union bound over $k$, we
control the probability of $E_{1}$ failing for these $k$.

For $k\geq k_{*}-2$, integrating the differential representation for
$\mathfrak{D}_{t}$, we have that for $(u,s)\in\operatorname{CP}_{k}$

$\displaystyle\mathfrak{D}_{u}-\mathfrak{D}_{s}=X_{u}-X_{s}+\int_{s}^{u}\biggl{(}\frac{\mathbf{1}\left\\{{t\leq
T_{*}}\right\\}}{\mathfrak{D}_{t}}\biggr{)}dt+\int_{s}^{u}\biggl{(}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{t}}{T_{+}-t}\biggr{)}dt.$

On the event $\mathscr{P}_{j}^{\prime}(\theta)$, we may bound
$\mathfrak{D}_{t}\geq\sqrt{\tfrac{8}{\beta}}(\log k_{5})$ (which is due to the
concave barrier in $\mathscr{P}_{j}^{\prime}(\theta)$ being shifted by a
constant factor from the conditioning (4.42)), for all $t\in(s,u)$ for which
$t\leq T_{*}$. To the bridge generator integral, we add and subtract
$\mathfrak{D}_{T_{+}-k+2}$ in the following way

$\int_{s}^{u}\biggl{(}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{t}}{T_{+}-t}\biggr{)}dt=\int_{s}^{u}\biggl{(}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{T_{+}-k+2}}{T_{+}-t}\biggr{)}dt+\int_{s}^{u}\biggl{(}\frac{\mathfrak{D}_{T_{+}-k+2}-\mathfrak{D}_{t}}{T_{+}-t}\biggr{)}dt.$

The first part we control in the same way as above. As for the second, we may
bound it above to create the following implicit bound

$\displaystyle\sup_{(u,s)\in\operatorname{CP}_{k}}\frac{|\mathfrak{D}_{u}-\mathfrak{D}_{s}|}{|u-s|^{1/4}}$
$\displaystyle\quad\leq\sup_{(u,s)\in\operatorname{CP}_{k}}\frac{|X_{u}-X_{s}|}{|u-s|^{1/4}}+(\mathfrak{u}+C_{\beta})+\frac{C}{k_{*}-4}\biggl{(}\max_{\kappa=k_{*}-2,\dots,k_{*}+2}\sup_{(u,s)\in\operatorname{CP}_{\kappa}}\frac{|\mathfrak{D}_{u}-\mathfrak{D}_{s}|}{|u-s|^{1/4}}\biggr{)}.$

Here $C_{\beta}$ controls the Bessel generator term and $C$ is an absolute
constant. Hence, on taking maxima over both sides, we conclude for all $k_{*}$
larger than an absolute constant

$\sup_{(u,s)\in\operatorname{CP}_{k}}\frac{|\mathfrak{D}_{u}-\mathfrak{D}_{s}|}{|u-s|^{1/4}}\leq
2\biggl{(}\max_{\kappa=k_{*}-2,\dots,k_{*}+2}\sup_{(u,s)\in\operatorname{CP}_{\kappa}}\frac{|{X}_{u}-{X}_{s}|}{|u-s|^{1/4}}+\mathfrak{u}+C_{\beta}\biggr{)}.$

Using (4.49) and (4.47), we conclude that there is some
$C_{\beta},c_{\beta}>0$ such that for all sufficiently small $\theta_{0}$ and
$\widehat{\delta}$,

${\mathbb{Q}}(E_{1}^{c}\cap
E_{0}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathscr{H}_{T_{-}})\leq
C_{\beta}e^{-c_{\beta}\widehat{\delta}^{2}(\log(1/\theta_{0}))^{1.01}}.$
(4.50)

##### Step 4: Finding a good event on which the imaginary part can not
explode.

This will turn out to be the most probabilistically expensive part. Let
$E_{2}$ be the event, for $C_{\beta},\delta_{\beta}$ to be determined,

$\mathfrak{X}_{t}-\mathfrak{X}_{s}\leq\left(1+\frac{2}{\beta}-3\widehat{\delta}\right)(t-s)-\delta_{\beta}\bigl{(}\log(\theta_{0})\bigr{)}+\log_{+}(T_{+}-t),\quad\forall\,T_{+}\geq
t\geq s\geq T_{-}.$ (4.51)

As we will work on the event $E_{1}$ it suffices to control the above on
integer–valued points, hence it instead suffices to bound the probability

$\displaystyle\mathfrak{X}_{t}-\mathfrak{X}_{s}$
$\displaystyle\leq\left(1+\frac{2}{\beta}-3\widehat{\delta}\right)(t-s)-(\delta_{\beta}-2\widehat{\delta})\bigl{(}\log(\theta_{0})\bigr{)}+(1-2\widehat{\delta})\log_{+}(T_{+}-t),$
$\displaystyle\forall\ T_{+}\geq t\geq s\geq T_{-}\quad\text{for which}\quad
t,s\in{\mathbb{Z}}.$

We can furthermore formulate a sufficient bound in terms of $\mathfrak{D}$
using that $|\ell^{\prime}(t)-\sqrt{\tfrac{8}{\beta}}|\leq\widehat{\delta}$
(which holds provided $k_{5}$ is chosen sufficiently large with respect to
$\widehat{\delta}$), so that

$\displaystyle\mathfrak{D}_{t}-\mathfrak{D}_{s}$
$\displaystyle\leq\left(1+\frac{2}{\beta}+\sqrt{\frac{8}{\beta}}-4\widehat{\delta}\right)(t-s)-(\delta_{\beta}-2\widehat{\delta})\bigl{(}\log(\theta_{0})\bigr{)}+(1-2\widehat{\delta})\log_{+}(T_{+}-t),$
(4.52) $\displaystyle\forall\ T_{+}\geq t\geq s\geq T_{-}\quad\text{for
which}\quad t,s\in{\mathbb{Z}}.$

If $s\geq T_{*}$, then as in the lead-up to (4.47), conditioning on the value
of $\mathfrak{D}_{T_{*}}$ the process
$(\mathfrak{D}_{t}-\mathfrak{D}_{T_{*}}:t\in[T_{*},T_{+}])$ is a
speed–$({\tfrac{4}{\beta}})$ Brownian bridge under ${\mathbb{Q}}$ with slope
at most $\mathfrak{u}$. Hence we have a tail bound that fany $t\geq s$ with
$s\geq T_{*}$

$\displaystyle{\mathbb{Q}}\left(\bigl{\\{}\mathfrak{D}_{t}-\mathfrak{D}_{s}>\delta+\tfrac{\widehat{\delta}}{3}(t-s)\bigr{\\}}\cap
E_{0}~{}\middle|~{}\mathscr{H}_{T_{-}}\right)\leq\exp\left(-\frac{\beta\delta^{2}}{8(t-s)}\right).$
(4.53)

Using this, we have that the portion of (4.52) for which $t,s$ with $T_{*}\leq
s\leq t\leq T_{+}$ can be controlled by $\widehat{\delta}\log(1/\theta_{0})$,
which is less than the bound stated in (4.52), with probability
$1-C_{\beta}e^{-c_{\beta}\widehat{\delta}^{2}(\log(1/\theta_{0}))^{2}}$ for
all $\widehat{\delta},\theta_{0}>0$ sufficiently small. Hence, we may reduce
the problem to showing

$\displaystyle\mathfrak{D}_{t}-\mathfrak{D}_{s}$
$\displaystyle\leq\left(1+\frac{2}{\beta}+\sqrt{\frac{8}{\beta}}-4\widehat{\delta}\right)(t-s)-(\delta_{\beta}-3\widehat{\delta})\bigl{(}\log(\theta_{0})\bigr{)}+(1-2\widehat{\delta})\log_{+}(T_{+}-t),$
(4.54) $\displaystyle\forall\ T_{*}\geq t\geq s\geq T_{-}\quad\text{for
which}\quad t,s\in{\mathbb{Z}}.$

For all $s\leq t\leq T_{*}$, we begin by recalling that the Bessel bridge SDE
has strong solutions:

$d\mathfrak{D}_{t}=dX_{t}+\biggl{(}\frac{\mathbf{1}\left\\{{t\leq
T_{*}}\right\\}}{\mathfrak{D}_{t}}+\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{t}}{T_{+}-t}\biggr{)}dt,\quad\mathfrak{D}_{T_{-}}+\ell(T_{-})=-\mathfrak{U}^{j}_{T_{-}}(0).$

For all $k_{5}$ sufficiently large (with respect to $\widehat{\delta}$) on
$\mathscr{P}_{j}^{\prime}(0)$, we may bound above the Bessel generator term by
$\widehat{\delta}$. The slope we bound on $E_{0}\cap
E_{1}\cap\mathscr{P}_{j}^{\prime}(0)$ by comparing $t$ to its (integer)
ceiling

$\biggl{|}\frac{\mathfrak{D}_{T_{+}}-\mathfrak{D}_{t}}{T_{+}-t}\biggr{|}\leq\frac{2\widehat{\delta}}{3}\frac{\log(1/\theta_{0})^{0.51}}{(T_{+}-t)^{0.51}}.$
(4.55)

When $T_{+}-t\geq\log(1/\theta_{0})$ we can just as well bound the above by
$\frac{2\widehat{\delta}}{3}$. We conclude that we have the bound on
$E_{0}\cap E_{1}\cap\mathscr{P}_{j}^{\prime}(0)$

$\mathfrak{D}_{t}-\mathfrak{D}_{s}\leq
X_{t}-X_{s}+\frac{5\widehat{\delta}}{3}(t-s)+\frac{5\widehat{\delta}}{3}\log(1/\theta_{0})\quad\text{for
all}\quad T_{-}\leq s\leq t\leq T_{*}.$ (4.56)

Thus for any $t\geq s$ with $s\leq T_{*}$ we have the bound, using the
probability that Brownian motion hits a line,

$\displaystyle{\mathbb{Q}}\left(\exists~{}s\in[T_{-},t]:\mathfrak{D}_{t}-\mathfrak{D}_{s}>\delta+\biggl{(}1+\frac{2}{\beta}+\sqrt{\frac{8}{\beta}}-4\widehat{\delta}\biggr{)}(t-s)~{}\bigg{|}~{}\mathscr{H}_{T_{-}}\right)$
$\displaystyle\leq\exp\left(-\frac{\beta}{2}\biggl{(}1+\frac{2}{\beta}+\sqrt{\frac{8}{\beta}}-\frac{17\widehat{\delta}}{3}\biggr{)}\biggl{(}\delta-\frac{5\widehat{\delta}}{3}\log(1/\theta_{0})\biggr{)}\right).$

We pick a $\delta_{\beta}\in(0,1)$ and an $\eta_{\beta}$ so that

$\frac{\beta}{2}\biggl{(}1+\frac{2}{\beta}+\sqrt{\frac{8}{\beta}}\biggr{)}\delta_{\beta}>\eta_{\beta}>1.$
(4.57)

We apply the tail bound just above with
$\delta=(\delta_{\beta}-3\widehat{\delta})\bigl{(}\log(1/(\theta_{0}))\bigr{)}+\bigl{(}1-2\widehat{\delta}\bigr{)}\log_{+}(T_{+}-t),$
from which it follows that for all $\widehat{\delta}$ sufficiently small (as a
function of $\beta$),

$\displaystyle{\mathbb{Q}}\left(\biggl{\\{}\exists~{}s\in[T_{-},t]:\text{
\eqref{eq:tlv5} fails }\biggr{\\}}\cap E_{0}\cap
E_{1}\cap\mathscr{P}_{j}^{\prime}(0)~{}\bigg{|}~{}\mathscr{H}_{T_{-}}\right)\leq(T_{+}-t)^{-(1/2)\log(1/\theta_{0})}\theta_{0}^{\eta_{\beta}+\widehat{\delta}}$

Summing in $t$, it is seen that for $\widehat{\delta}$ sufficiently small and
$\theta_{0}$ sufficiently small to absorb the constants,

${\mathbb{Q}}((E_{2})^{c}\cap E_{0}\cap
E_{1}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathscr{H}_{T_{-}})\leq\theta_{0}^{\eta_{\beta}+\widehat{\delta}/2}.$
(4.58)

##### Step 5: A tail bound for the change in the imaginary part.

We estimate the imaginary part of

$\Delta_{t}=\Delta_{t}(\theta_{0})=\Im(\mathfrak{L}_{t}(\theta_{0})-\mathfrak{L}_{t}(0)).$

This satisfies the SDE

$d\Delta_{t}=\theta_{0}e^{t}k_{1}^{-1}dt+\sqrt{\tfrac{4}{\beta}}\Im\biggl{(}(e^{i\Delta_{t}}-1)e^{i\Im\mathfrak{L}_{t}(0)}d\mathfrak{W}_{t}^{j}\biggr{)},$

which has almost surely non-negative solutions. Then we express

$d\Delta_{t}=\theta_{0}e^{t}k_{1}^{-1}dt+\Delta_{t}d\mathfrak{X}_{t}+\xi_{1}(\Delta_{t})d\mathfrak{X}_{t}+\xi_{2}(\Delta_{t})d\mathfrak{B}_{t},$

where $\xi_{1}(x)$ and $\xi_{2}(x)$ are bounded by $C_{\beta}x^{2}.$

Let $M_{t}=\exp\left(\mathfrak{X}_{t}-\tfrac{2}{\beta}t\right).$ Then we have
from Itô’s Lemma

$d\biggl{(}\frac{\Delta_{t}}{M_{t}}\biggr{)}=\frac{1}{M_{t}}\biggl{(}\theta_{0}e^{t}k_{1}^{-1}dt+\xi_{1}(\Delta_{t})(d\mathfrak{X}_{t}-\tfrac{2}{\beta}dt)+\xi_{2}(\Delta_{t})d\mathfrak{B}_{t}\biggr{)}.$
(4.59)

This we can integrate to conclude for $t\in[T_{-},T_{+}],$

$\Delta_{t}=\frac{M_{t}}{M_{T_{-}}}\Delta_{T_{-}}+\int_{T_{-}}^{t}\frac{M_{t}}{M_{s}}\bigl{(}\theta_{0}e^{s}k_{1}^{-1}ds+\xi_{1}(\Delta_{s})(d\mathfrak{X}_{s}-\tfrac{2}{\beta}ds)+\xi_{2}(\Delta_{s})d\mathfrak{B}_{s}\bigr{)}.$
(4.60)

Now by definition we have

$\frac{M_{t}}{M_{s}}=\exp\left(\mathfrak{X}_{t}-\mathfrak{X}_{s}-\frac{2}{\beta}(t-s)\right).$
(4.61)

Recalling (4.51), on the event $E_{0}\cap E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)$,

$\frac{M_{t}}{M_{s}}\leq\exp\left(\bigl{(}1-3\widehat{\delta}\bigr{)}(t-s)-\delta_{\beta}\bigl{(}\log(\theta_{0})\bigr{)}+\log_{+}(T_{+}-t)\right).$
(4.62)

Let

$L_{t}=\int_{T_{-}}^{t}\frac{M_{t}}{M_{s}}\theta_{0}e^{s}k_{1}^{-1}ds,\quad\text{for
any}\quad t\in[T_{-},T_{+}].$

On the event $E_{0}\cap E_{1}\cap E_{2}\cap\mathscr{P}_{j}^{\prime}(0),$ we
have from (4.62) the bound

$|L_{t}|\leq\widehat{\delta}^{-1}(\theta_{0})^{1-\delta_{\beta}}e^{-(T_{+}-t)+\log_{+}(T_{+}-t)}\eqqcolon(\theta_{0})^{1-\delta_{\beta}}\mathcal{W}(t).$
(4.63)

for $t\in[T_{-},T_{+}].$

Let $\tau$ be the first time $s$ greater than $T_{-}$ that

$|\Delta_{s}-L_{s}-M_{s}M_{T_{-}}^{-1}\Delta_{T_{-}}|\geq(\theta_{0})^{1-\delta_{\beta}}\bigl{(}\mathcal{W}(t)+(\log
k_{1})^{-\alpha/2}\bigr{)},$

and note that from (4.63), for all $t\in[T_{-},T_{+}],$ and on the event
$E_{0}\cap E_{1}\cap E_{2}\cap\mathscr{P}_{j}^{\prime}(0)$ and the conditions
on $\alpha,\theta_{0}$ ((4.40) and (4.39)),

$\displaystyle\Delta_{t\wedge\tau}=|\Delta_{t\wedge\tau}|$
$\displaystyle\leq|\Delta_{t\wedge\tau}-L_{t\wedge\tau}-M_{t\wedge\tau}M_{T_{-}}^{-1}\Delta_{T_{-}}|+|M_{t\wedge\tau}M_{T_{-}}^{-1}\Delta_{T_{-}}|+|L_{t\wedge\tau}|$
(4.64) $\displaystyle\leq
2(\theta_{0})^{1-\delta_{\beta}}\bigl{(}\mathcal{W}(t\wedge\tau)+(\log
k_{1})^{-\alpha/2}\bigr{)}.$

Returning to the SDE for $\Delta,$ we now write

$d\Delta_{t}=\theta_{0}e^{t}k_{1}^{-1}dt+\Delta_{t}dU_{t}\quad\text{where}\quad
dU_{t}=d\mathfrak{X}_{t}+\tfrac{\xi_{1}(\Delta_{t})}{\Delta_{t}}d\mathfrak{X}_{t}+\tfrac{\xi_{2}(\Delta_{t})}{\Delta_{t}}d\mathfrak{B}_{t},$

Letting $N_{t}=\exp\bigl{(}U_{t}-\tfrac{1}{2}\langle U_{t}\rangle\bigr{)}$, we
therefore have

$\Delta_{t}-L_{t}-\frac{M_{t}}{M_{T_{-}}}\Delta_{T_{-}}=\biggl{(}\frac{N_{t}}{N_{T_{-}}}-\frac{M_{t}}{M_{T_{-}}}\biggr{)}\Delta_{T_{-}}+\int_{T_{-}}^{t}\biggl{(}\frac{N_{t}}{N_{s}}-\frac{M_{t}}{M_{s}}\biggr{)}\bigl{(}\theta_{0}e^{s}k_{1}^{-1}ds\bigr{)}.$
(4.65)

This essentially reduces the problem to an estimate on the ratios of
integrating factors that holds up to the stopping time. First we observe that
we have the representation

$\log\biggl{(}\frac{N_{t}}{N_{T_{-}}}\frac{M_{T_{-}}}{M_{t}}\biggr{)}=\int_{T_{-}}^{t}\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}d\mathfrak{X}_{u}+\tfrac{\xi_{2}(\Delta_{u})}{\Delta_{u}}d\mathfrak{B}_{u}-\frac{2}{\beta}\int_{T_{-}}^{t}\bigl{(}2\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}+\bigl{(}\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}\bigr{)}^{2}+\bigl{(}\tfrac{\xi_{2}(\Delta_{u})}{\Delta_{u}}\bigr{)}^{2}\bigr{)}\,du$

We bound the right hand side uniformly over $T_{-}\leq t\leq\tau$. There are
three types of terms to control: the finite variation terms in the second
integral $(i)$, the martingale terms in the first integral $(ii)$, and the
finite variation terms in the first integral $(iii)$. Then for the first
terms, using the bound on $\Delta$ in (4.64):

$(i)\coloneqq\int_{T_{-}}^{T_{+}\wedge\tau}\bigl{|}2\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}+\bigl{(}\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}\bigr{)}^{2}+\bigl{(}\tfrac{\xi_{2}(\Delta_{u})}{\Delta_{u}}\bigr{)}^{2}\bigr{|}du\leq
C(\beta)(\theta_{0})^{1-\delta_{\beta}}$

For the second terms, we need a stochastic control, and we have by bounding
the quadratic variation for some constant

$\displaystyle{\mathbb{Q}}\left(\biggl{\\{}\max_{s\leq
T_{+}\wedge\tau}\biggl{|}\int_{T_{-}}^{s}\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}d{X}_{u}+\tfrac{\xi_{2}(\Delta_{u})}{\Delta_{u}}d\mathfrak{B}_{u}\biggr{|}>x\biggr{\\}}\cap
E_{0}\cap E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)~{}\middle|~{}\mathscr{H}_{T_{-}}\right)$
$\displaystyle\leq\exp\biggl{(}-\frac{x^{2}}{C(\beta,\widehat{\delta})(\theta_{0})^{4(1-\delta_{\beta})}}\biggr{)}.$

In particular, we may assume with probability
$1-e^{O(\theta_{0}^{-2(1-\delta_{\beta})})},$ that

$(ii)\coloneqq\max_{s\leq
t\wedge\tau}\biggl{|}\int_{T_{-}}^{s}\tfrac{\xi_{1}(\Delta_{u})}{\Delta_{u}}d{X}_{u}+\tfrac{\xi_{2}(\Delta_{u})}{\Delta_{u}}d\mathfrak{B}_{u}\biggr{|}\leq\theta_{0}^{1-\delta_{\beta}}.$
(4.66)

Finally for the third terms,

$(iii)\coloneqq\int_{T_{-}}^{T_{+}\wedge\tau}\Delta_{s}\biggl{(}\biggl{|}\frac{\mathbf{1}\left\\{{s\leq
T_{*}}\right\\}}{\ell(s)+\mathfrak{X}_{s}}\biggr{|}+\biggl{|}\frac{\mathfrak{X}_{T_{+}}-\mathfrak{X}_{s}}{T_{+}-s}\biggr{|}\biggr{)}ds$

On the event $\mathscr{P}_{j}^{\prime}(0),$

$\biggl{|}\frac{\mathbf{1}\left\\{{s\leq
T_{*}}\right\\}}{\ell(s)+\mathfrak{X}_{s}}\biggr{|}\leq 2.$

On the event $E_{0}\cap E_{1}\cap\mathscr{P}_{j}^{\prime}(0),$ using (4.55)
and the control for $\ell^{\prime}$, we have

$\biggl{|}\frac{\mathfrak{X}_{T_{+}}-\mathfrak{X}_{s}}{T_{+}-s}\biggr{|}\leq\widehat{\delta}\log(1/\theta_{0})$
(4.67)

Hence applying these bounds, we have for some $C(\beta,\widehat{\delta})$
sufficiently large

$(iii)\leq
C(\beta,\widehat{\delta})\theta_{0}^{1-\delta_{\beta}}\log(1/\theta_{0}).$

Combining all of these, we conclude that for some $C(\beta,\widehat{\delta})$
sufficiently large

$\max_{T_{-}\leq s\leq t\leq
T_{+}\wedge\tau}\biggl{|}\log\biggl{(}\frac{N_{t}}{N_{s}}\frac{M_{s}}{M_{t}}\biggr{)}\biggr{|}\leq
C(\beta,\widehat{\delta})\theta_{0}^{1-\delta_{\beta}}\log(1/\theta_{0}).$
(4.68)

Hence we conclude from (4.65) that for $t\leq\tau$, for some
$C(\beta,\widehat{\delta})$ and all $\theta_{0}$ sufficiently small (depending
on $\beta,\widehat{\delta}$),

$|\Delta_{t}-L_{t}-\frac{M_{t}}{M_{T_{-}}}\Delta_{T_{-}}|\leq\biggl{(}\exp\bigl{(}C\theta_{0}^{1-\delta_{\beta}}\log(1/\theta_{0})\bigr{)}-1\biggr{)}\cdot\biggl{(}\frac{M_{t}}{M_{T_{-}}}\Delta_{T_{-}}+\int_{T_{-}}^{t}\frac{M_{t}}{M_{s}}\bigl{(}\theta_{0}e^{s}k_{1}^{-1}ds\bigr{)}\biggr{)}.$

We conclude as in (4.64) that for some $C=C(\beta,\widehat{\delta})$,

$|\Delta_{T_{+}\wedge\tau}-L_{T_{+}\wedge\tau}-\frac{M_{T_{+}\wedge\tau}}{M_{T_{-}}}\Delta_{T_{-}}|\leq
C\theta_{0}^{2(1-\delta_{\beta})}\log(1/\theta_{0})\bigl{(}\mathcal{W}(t)+(\log
k_{1})^{-\alpha/2}\bigr{)}.$

By the definition of $\tau,$ we conclude that on the events considered,
$E_{0}\cap E_{1}\cap E_{2}\cap\mathscr{P}_{j}^{\prime}(0)$, as well as on the
event in which (4.66) holds, that $\tau>T_{+}$, and hence (4.64) holds for all
$t\in[T_{-},T_{+}].$ In summary, we conclude there is an $\eta_{\beta}$ (see
(4.57)), a $\delta_{\beta}\in(0,1)$ a $C_{\beta}>0$ and an $\epsilon>0$ so
that for all $\theta_{0}$ sufficiently small,

$\displaystyle{\mathbb{Q}}\Big{(}\\{\exists~{}t\in[T_{-},T_{+}]~{}:~{}$
$\displaystyle\qquad|\Delta_{t}|>C_{\beta}(\theta_{0})^{1-\delta_{\beta}}\bigl{(}e^{-(1-\widehat{\delta})(T_{+}-t)}+\log^{-\alpha/2}(k_{1})\bigr{)}\\}\cap
E_{0}\cap E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathscr{H}_{T_{-}}\Big{)}$
$\displaystyle\leq\theta_{0}^{\eta_{\beta}+\widehat{\delta}}.$ (4.69)

##### Step 6: Control for the real part.

Going forward, we work on the event $F_{\theta_{0}}$ on which

$\forall~{}t\in[T_{-},T_{+}]~{}:~{}|\Delta_{t}(\theta_{0})|\leq
C_{\beta}(\theta_{0})^{1-\delta_{\beta}}\bigl{(}e^{-(1-\widehat{\delta})(T_{+}-t)}+(\log
k_{1})^{-\alpha/2}\bigr{)},$

the probability of which is esimated in (4.2). The real part of
$\mathfrak{L}_{t}$ we will represent by

$\Delta_{t}^{r}(\theta)=\Re(\mathfrak{L}_{t}(\theta)-\mathfrak{L}_{t}(0)).$

Using the SDE for $\mathfrak{L}$ in (2.35)

$d\Delta_{t}^{r}=(\cos(\Delta_{t}(\theta))-1)d\mathfrak{X}_{t}-\sin(\Delta_{t}(\theta))d\mathfrak{B}_{t}.$

Since $0\leq\Delta_{t}(\theta)\leq\Delta_{t}(\theta_{0})$ we can estimate the
finite variation parts on $F_{\theta_{0}}\cap E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)$ using a similar analysis as in (4.68) by

$\int_{T_{-}}^{T_{+}\wedge\tau}\Delta_{s}^{2}\biggl{(}\biggl{|}\frac{\mathbf{1}\left\\{{s\leq
T_{*}}\right\\}}{\ell(s)+\mathfrak{X}_{s}}\biggr{|}+\biggl{|}\frac{\mathfrak{X}_{T_{+}}-\mathfrak{X}_{s}}{T_{+}-s}\biggr{|}\biggr{)}ds\leq
C(\beta,\widehat{\delta})\log(1/\theta_{0})|\theta_{0}|^{2-2\delta_{\beta}}.$
(4.70)

The quadratic variation is dominated by on the event $F_{\theta_{0}}$ by, for
some sufficiently large $C_{\beta},$

$\langle\Delta_{T_{+}}^{r}(\theta)\rangle\leq
C_{\beta}\int_{T_{-}}^{T_{+}}\Delta_{t}^{2}(\theta_{0})dt\leq
C_{\beta}^{2}\bigl{(}\theta_{0}\bigr{)}^{2-2\delta_{\beta}}.$

Therefore, we have a tail bound that for some $C_{\beta}$ sufficiently large
and for all $x>0,$

${\mathbb{Q}}\left(\\{|\Delta_{T_{+}}^{r}(\theta_{0})|>C_{\beta}(1+x)|\theta_{0}|^{1-\delta_{\beta}}\\}\cap
F_{\theta_{0}}\cap E_{0}\cap E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathscr{H}_{T_{-}}\right)\leq
e^{-x^{2}}$

Thus taking $x=\theta_{0}^{-\varepsilon}$ for $\varepsilon<1-\delta_{\beta}$
we conclude that

${\mathbb{Q}}\bigl{(}\mathscr{O}_{*}^{c}\cap F_{\theta_{0}}\cap E_{0}\cap
E_{1}\cap
E_{2}\cap\mathscr{P}_{j}^{\prime}(0)~{}|~{}\mathscr{H}_{T_{-}}\bigr{)}\leq
e^{-\theta_{0}^{-\delta}},$

which finally concludes (4.45). ∎

##  5 The diffusion approximation

In this section, we prove Proposition 2.12. The proof is given by a series of
short lemmas. A summary proof is given in the penultimate section. In the
final section, we develop a general tail bound which we use at multiple points
in the development.

It is convenient in this section to work conditionally on the event
$\mathscr{T}_{n_{1}^{+}}$ from (2.7). We let, for the event
$\mathscr{T}_{n_{1}^{+}}$ and $\sigma$–filtration $\mathscr{F}_{n_{1}^{+}}$,

$\mathbb{P}_{M}(\cdot)\coloneqq\mathbb{P}(\cdot~{}|~{}\mathscr{T}_{n_{1}^{+}},\mathscr{F}_{n_{1}^{+}},(\Gamma_{j}:j>n_{1}^{+})),$

and let $\mathbb{E}_{M}$ denote the associated expectation. Under the law
$\mathbb{P}_{M},$ $(X_{j},Y_{j})$ (see (2.2)) are no longer independent of one
another for $j\geq n_{1}^{+}$. They do however still satisfy uniform Gaussian
integrability, in that

$\mathbb{E}_{M}\left[\exp(\lambda
F(X_{j},Y_{j}))\right]\leq\exp\left(\lambda\mathbb{E}_{M}\left[F(X_{j},Y_{j})\right]+{\lambda^{2}\|\nabla
F(X_{j},Y_{j})\|^{2}_{\text{L}^{\infty}(\mathbb{P}_{M})}}\right),\quad\text{
for all }\lambda\in{\mathbb{R}},$ (5.1)

for all Lipschitz $F$ on the (convex) support of $(X_{j},Y_{j})$. We will use
this subgaussian concentration for the family of functions that appear in the
definitions of $\log\Phi^{*}_{k}(\theta)$, and in particular to control the
linearization error.

###  5.1 Locally linear processes

In a similar fashion to [CMN18], we introduce a process which in short windows
of $k$ evolves linearly. Let $\varkappa$ be a parameter, to be chosen later as
a power of $n$, which will be the block length within which $\psi_{k}$ will
evolve linearly. Define a new recurrence

$\displaystyle\lambda_{k+1}(\theta)=\lambda_{k}(\theta)+i\theta+\frac{Z_{k}e^{i\Im\lambda^{*}_{k}(\theta)}}{\beta_{k}}\quad\text{for
all }k\geq{n_{1}^{+}},\quad\text{where}\quad Z_{k}=X_{k}+iY_{k},$ (5.2)
$\displaystyle\lambda_{{n_{1}^{+}}}(\theta)=2\log\Phi^{*}_{{n_{1}^{+}}}(\theta),\;\lambda^{*}_{k}(\theta)=\lambda_{{n_{1}^{+}}+n^{*}(k)}(\theta)+(k-n^{*}(k))\theta,\;\text{and}\;n^{*}(k)=\varkappa\lfloor\tfrac{k-{n_{1}^{+}}}{\varkappa}\rfloor\;\forall\,k\geq{n_{1}^{+}}.$

We begin with a simple observation that for $\varkappa$ sufficiently small
with respect to ${n_{1}^{+}},$ the difference between $\lambda_{k}$ and
$\lambda_{k}^{*}$ can be controlled.

######  Lemma 5.1.

For all $C>0$ there is constant $D>0$ so that for all ${n_{1}^{+}}$
sufficiently large (depending on $\beta$ and $M$),

$\mathbb{P}_{M}\left[\max_{{n_{1}^{+}}\leq
k<n}|\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta)|\geq
D\sqrt{\tfrac{\varkappa\log n}{\beta{n_{1}^{+}}}}\right]\leq n^{-C}.$

###### Proof.

We show the proof for ${n_{1}^{+}}\leq k<{n_{1}^{+}}+\varkappa.$ For larger
$k,$ we have the same estimate (and indeed it only improves). We have that

$\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta)=\sum_{j={n_{1}^{+}}}^{k-1}\frac{Z_{j}e^{i\Im\lambda^{*}_{j}(\theta)}}{\beta_{j}}.$

Under $\mathbb{P}$, this is Gaussian with variance
$\tfrac{4}{\beta}\sum_{j={n_{1}^{+}}}^{k-1}\frac{1}{j+1}\leq\tfrac{4\varkappa}{\beta{n_{1}^{+}}}.$
Hence there is an absolute constant so that for all $t\geq 0$ and all $\theta$

$\mathbb{P}_{M}\left[\max_{{n_{1}^{+}}\leq
k<n_{1}^{+}+\varkappa}|\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta)|\geq
t\right]\leq\frac{2}{\mathbb{P}(\mathscr{T}_{{n_{1}^{+}}})}\exp\left(-\beta(n_{1}^{+}t)^{2}/(C\varkappa)\right).$

In particular for $t=D\sqrt{\tfrac{\varkappa\log n}{\beta{n_{1}^{+}}}}$ with
$D$ sufficiently large, the claim follows. ∎

We turn to comparing the differences $R_{k}+i\Delta_{k}\coloneqq
2\log\Phi^{*}_{k}-\lambda_{k}$ (for real-valued $R$ and $\Delta$). We begin by
observing that the difference satisfies a recurrence for $k\geq{n_{1}^{+}}$:

$\displaystyle R_{k+1}(\theta)+i\Delta_{k+1}(\theta)$
$\displaystyle=R_{k}(\theta)+i\Delta_{k}(\theta)+2\left(-\log(1-\gamma_{k}e^{i\psi_{k}(\theta)})-\left(\tfrac{Z_{k}e^{i\Im\lambda^{*}_{k}(\theta)}}{\beta_{k}}\right)\right)$
(5.3)
$\displaystyle=R_{k}(\theta)+i\Delta_{k}(\theta)+2\left(-\log(1-\tilde{\gamma_{k}}e^{i\Delta_{k}(\theta)+i\Im(\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta))})-\left(\tfrac{\tilde{Z_{k}}}{\beta_{k}}\right)\right),$

where $\left\\{(\tilde{\gamma_{k}},\tilde{Z_{k}})\right\\}$ have the same law
as $\left\\{(\gamma_{k},Z_{k})\right\\}.$ The proof of the following lemma
uses calculus computations contained in Section 7.

######  Lemma 5.2.

Suppose $\varkappa\geq\sqrt{{n_{1}^{+}}}.$ For all $\delta>0$ and all $C>0$.

$\mathbb{P}_{M}\biggl{[}\max_{{n_{1}^{+}}\leq k\leq n}|R_{k}+i\Delta_{k}|\geq
n^{\delta}\sqrt{\varkappa/{n_{1}^{+}}}\biggr{]}\leq n^{-C}$

for all $n$ sufficiently large.

###### Proof.

We show the bound for the imaginary part. We can express

$\mathbb{E}_{M}\left[e^{\mu(\Delta_{k+1}-\Delta_{k})}\middle|\mathscr{F}_{k}\right]=\mathbb{E}_{M}\left[e^{\mu
F(X_{j},Y_{j})}\right],$

where in the notation of Lemma 7.1,

$F(x,y)=\Re\left\\{i\log(1-u(x+iy)e^{i\Delta_{k}(\theta)+i\Im(\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta))})+i(x+iy)/\beta_{k}\right\\}.$

Hence by (5.1) and Lemma 7.1, there is a constant $C>0$ so that for any
$k\geq{n_{1}^{+}},$ and any $\mu\in{\mathbb{R}}$

$\mathbb{E}_{M}\left[e^{\mu\Delta_{k+1}}\middle|\mathscr{F}_{k}\right]\leq\exp\left(\mu\Delta_{k}+\frac{C\mu^{2}|e^{i\Delta_{k}(\theta)+i\Im(\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta))}-1|^{2}}{\beta_{k}^{2}}+\frac{C\mu^{2}\log
k}{\beta_{k}^{3}}\right).$

Let $T$ be the first $k\geq{n_{1}^{+}}$ such that
$|\Im(\lambda_{k}(\theta)-\lambda_{k}^{*}(\theta))|$ is larger than
$D\sqrt{\tfrac{\varkappa\log n}{\beta{n_{1}^{+}}}}$ for some $D>\sqrt{\beta}.$
We have supposed that $\varkappa\geq\sqrt{{n_{1}^{+}}},$ and therefore there
is an absolute constant $C>0$ so that for any $k\geq{n_{1}^{+}}$ and any
$\mu\in{\mathbb{R}},$

$\mathbb{E}_{M}\left[e^{\mu\Delta^{T}_{k+1}}\middle|\mathscr{F}_{k}\right]\leq\exp\left(\mu\Delta^{T}_{k}+\frac{C\mu^{2}((\Delta^{T}_{k})^{2}+D^{2}\tfrac{\varkappa\log
n}{\beta{n_{1}^{+}}})}{\beta k}\right).$

In preparation to use Proposition 5.6, we observe from (5.3) and Lemma 7.1
that there is an absolute constant $C>0$ so that for all ${n_{1}^{+}}$
sufficiently large,

$\Delta^{T}_{k+1}(\theta)-\Delta^{T}_{k}(\theta)\leq
C\Delta^{T}_{k}\sqrt{\frac{\log({n_{1}^{+}})}{\beta{n_{1}^{+}}}}+CD\sqrt{\frac{\varkappa\log
n}{\beta{n_{1}^{+}}}}.$

Hence by Proposition 5.6, there is an absolute constant $C>0$ so that for all
$x\geq C\log(n/{n_{1}^{+}})/\beta$ $+C,$

$\mathbb{P}_{M}\left[\max_{{n_{1}^{+}}\leq k\leq n}\Delta^{T}_{k}\geq
xD\sqrt{\varkappa\log(n)/(\beta{n_{1}^{+}})}\right]\leq\exp\left(-\frac{1}{C}\frac{\log(x)^{2}}{\log(n/{n_{1}^{+}})}\right).$

The same bound holds for $-\Delta_{k}^{T},$ and therefore by Lemma 5.1, the
lemma follows.

To control the real part $R_{k}$, we again use Lemma 7.1, although there is no
longer a need for the ladder. We suppress the details. ∎

###  5.2 Band–resampled approximation

Recall from (2.2), that $Z_{\ell}=\sqrt{E_{\ell}}e^{i\Theta_{\ell}}$ is a
complex Gaussian for each $\ell.$ For any $r\in{\mathbb{N}},$ define
$k_{r}={n_{1}^{+}}+r\varkappa.$ The family of Gaussians
$\left\\{Z_{\ell}:k_{r-1}\leq\ell<k_{r}\right\\}$ are i.i.d. Hence, we can
represent for any $r\in{\mathbb{N}}$, these Gaussians through their discrete
Fourier transform:

$Z_{\ell+k_{r-1}}=\frac{1}{\sqrt{\varkappa}}\sum_{p=1}^{\varkappa}e(\tfrac{-p\ell}{\varkappa})\hat{Z}_{p}^{(r)},\quad\text{for
}0\leq\ell\leq\varkappa-1$ (5.4)

where $e(x)=e^{2\pi ix}$ and $\left\\{\hat{Z}_{p}^{(r)}:r\in{\mathbb{N}},1\leq
p\leq\varkappa\right\\}$ are another family of i.i.d. complex Gaussians.

We shall estimate the effect on the recurrence $\lambda_{k}(\theta)$ wherein
we resample some of the $\\{\hat{Z}_{p}\\}$ corresponding to modes that are
far from $\theta.$ Let $\omega\leq\varkappa$ be a positive integer parameter,
which will be the bandwidth. For a fixed $j\in{\mathcal{D}}_{n/k_{1}},$ define
for any $r\in{\mathbb{N}}$

$Z_{\ell+k_{r-1}}^{(j)}=\frac{1}{\sqrt{\varkappa}}\sum_{p=1}^{\varkappa}e(\tfrac{-p\ell}{\varkappa})(\hat{Z}_{p}^{(r)}+(\check{Z}_{p}^{(r,j)}-\hat{Z}_{p}^{(r)})\mathbf{1}\left\\{{|e(\tfrac{p}{\varkappa})-e(\tfrac{\theta_{j}}{2\pi})|>\tfrac{\omega}{\varkappa}}\right\\}),\quad\text{for
}0\leq\ell\leq\varkappa-1,$ (5.5)

where $\left\\{\check{Z}_{p}^{(r,j)}:r\in{\mathbb{N}},1\leq
p\leq\varkappa,j\in{\mathcal{D}}_{n/k_{1}}\right\\}$ is another family of
standard complex Gaussians.

We also define locally linear processes driven by these band–resampled
Gaussians:

$\displaystyle\lambda_{k+1}^{(j)}(\theta)=\lambda_{k}^{(j)}(\theta)+i\theta+2\frac{Z_{k}^{(j)}e^{i\Im\lambda^{*,(j)}_{k}(\theta)}}{\beta_{k}}\quad\text{for
all }k\geq{n_{1}^{+}},\quad\text{where}$ (5.6)
$\displaystyle\lambda_{{n_{1}^{+}}}^{(j)}(\theta)=\psi_{{n_{1}^{+}}}(\theta),\quad\text{
and
}\quad\lambda^{*,(j)}_{k}(\theta)=\lambda^{(j)}_{{n_{1}^{+}}+n^{*}(k)}(\theta)+(k-n^{*}(k))\theta\quad\text{for
all }k\geq{n_{1}^{+}}.$

We then define
$R^{(j)}_{k}(\theta)+i\Delta^{(j)}_{k}(\theta)\coloneqq\lambda_{k}(\theta)-\lambda_{k}^{(j)}(\theta).$

######  Lemma 5.3.

Suppose that $j\in\mathcal{D}_{n/k_{1}}$ and that $\theta\in{\mathbb{R}}$
satisfies

$|e(\tfrac{\theta_{j}}{2\pi})-e(\tfrac{\theta}{2\pi})|\leq\frac{\omega}{2\varkappa}.$

There is an absolute constant $C>0$ so that for any $r\in{\mathbb{N}}$ and all
$\mu\in{\mathbb{R}}$

$\mathbb{E}_{M}\left[e^{\mu\Delta_{k_{r}}^{(j)}}\middle|\mathscr{F}_{k_{r-1}}\right]\leq\exp\left(\mu\Delta_{k_{r-1}}^{(j)}+\frac{\mu^{2}C\varkappa|\Delta^{(j)}_{k_{r-1}}|^{2}}{\beta
k_{r-1}}+\frac{\mu^{2}C\varkappa}{\beta\omega k_{r-1}}\right),$

and so that for all $t\geq 0$

$\mathbb{P}_{M}\left[\max_{k_{r-1}\leq
k<k_{r}}|\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)}|\geq
t\middle|\mathscr{F}_{k_{r-1}}\right]\leq
2\exp\left(-\frac{t^{2}k_{r-1}\omega\beta}{C\varkappa(\omega|\Delta_{k_{r-1}}^{(j)}|^{2}+1)}\right).$

###### Proof.

For any $r\in{\mathbb{N}},$ we have that conditionally on
$\mathscr{F}_{k_{r-1}},$ $\left\\{\Delta_{k}^{(j)}:k_{r-1}\leq
k<k_{r}\right\\}$ are jointly Gaussian under $\mathbb{P}.$ Moreover, for such
$k,$ we can write

$\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)}=2\Im\sum_{\ell=0}^{k-k_{r-1}}\left\\{\frac{Z_{\ell+k_{r-1}}e^{i\ell\theta+i\lambda_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}-\frac{Z_{\ell+k_{r-1}}^{(j)}e^{i\ell\theta+i\lambda^{(j)}_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}\right\\}.$

We give an upper bound for the variance of
$\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)},$ which we do by separately bounding
the variance of

$A=2\Im\sum_{\ell=0}^{k-k_{r-1}}\left\\{\frac{Z_{\ell+k_{r-1}}e^{i\ell\theta+i\lambda_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}-\frac{Z_{\ell+k_{r-1}}^{(j)}e^{i\ell\theta+i\lambda_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}\right\\},$

and of

$B=2\Im\sum_{\ell=0}^{k-k_{r-1}}\left\\{\frac{Z_{\ell+k_{r-1}}^{(j)}e^{i\ell\theta+i\lambda_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}-\frac{Z_{\ell+k_{r-1}}^{(j)}e^{i\ell\theta+i\lambda^{(j)}_{k_{r-1}}}}{\beta_{\ell+k_{r-1}}}\right\\}.$

For the second one, we observe that

$\operatorname{Var}(B|\mathscr{F}_{k_{r-1}})\leq\frac{8\varkappa|\Delta^{(j)}_{k_{r-1}}|^{2}}{\beta
k_{r-1}}.$ (5.7)

The main work is to control the variance of $A.$ By rotation invariance, we
may drop the $e^{i\lambda_{k_{r-1}}}$ from both terms and write

$\displaystyle A$
$\displaystyle\overset{\mathscr{L}}{=}2\Im\sum_{\ell=0}^{k-k_{r-1}}\frac{e^{i\ell\theta}}{\sqrt{\varkappa}\beta_{\ell+k_{r-1}}}\sum_{p=1}^{\varkappa}e(\tfrac{-p\ell}{\varkappa})(\check{Z}_{p}^{(r,j)}-\hat{Z}_{p}^{(r)})\mathbf{1}\left\\{{|e(\tfrac{p}{\varkappa})-e(\tfrac{\theta_{j}}{2\pi})|>\tfrac{\omega}{\varkappa}}\right\\})$
$\displaystyle=2\Im\sum_{p=1}^{\varkappa}c_{p}(\check{Z}_{p}^{(r,j)}-\hat{Z}_{p}^{(r)})\mathbf{1}\left\\{{|e(\tfrac{p}{\varkappa})-e(\tfrac{\theta_{j}}{2\pi})|>\tfrac{\omega}{\varkappa}}\right\\}),$

where we have set

$c_{p}=\sum_{\ell=0}^{k-k_{r-1}}\frac{e(\ell(\tfrac{\theta}{2\pi}-\tfrac{p}{\varkappa}))}{\sqrt{\varkappa}\beta_{\ell+k_{r-1}}}.$

Moreover,

$\operatorname{Var}(A|\mathscr{F}_{k_{r-1}})=4\sum_{p=1}^{\varkappa}|c_{p}|^{2}\mathbf{1}\left\\{{|e(\tfrac{p}{\varkappa})-e(\tfrac{\theta_{j}}{2\pi})|>\tfrac{\omega}{\varkappa}}\right\\},$
(5.8)

and so it remains to estimate $|c_{p}|^{2}.$ Note that we have the simple
bound

$\left|{\textstyle\sum_{\ell=0}^{k-k_{r-1}}e(\ell(\tfrac{\theta_{j}}{2\pi}-\tfrac{p}{\varkappa}))}\right|\leq\frac{2}{|e(\tfrac{\theta_{j}}{2\pi})-e(\tfrac{p}{\varkappa})|}.$

Hence using summation-by-parts, we have that under the assumptions on $\theta$
and for $p$ for which
$|e(\tfrac{p}{\varkappa})-e(\tfrac{\theta_{j}}{2\pi})|>\tfrac{\omega}{\varkappa},$
there is an absolute constant $C>0$ so that

$|c_{p}|\leq\frac{1}{\sqrt{\varkappa}\beta_{k_{r-1}}}+\frac{2}{|e(\tfrac{\theta}{2\pi})-e(\tfrac{p}{\varkappa})|\sqrt{\varkappa}\beta_{k_{r-1}}}\leq\frac{C}{|e(\tfrac{\theta_{j}}{2\pi})-e(\tfrac{p}{\varkappa})|\sqrt{\varkappa}\beta_{k_{r-1}}}$

Thus turning to (5.8), we can bound for some absolute constant $C>0$

$\operatorname{Var}(A|\mathscr{F}_{k_{r-1}})\leq\frac{C}{\beta_{k_{r-1}}^{2}}+\sum_{p=\omega}^{\infty}\frac{C\varkappa}{p^{2}\beta_{k_{r-1}}^{2}}.$

Hence, we conclude from this equation and (5.7) that there is an absolute
constant $C>0$ so that

$\operatorname{Var}(\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)}|\mathscr{F}_{k_{r-1}})\leq\frac{C\varkappa|\Delta^{(j)}_{k_{r-1}}|^{2}}{\beta
k_{r-1}}+\frac{C\varkappa}{\beta\omega k_{r-1}}.$

As we condition on an event of probability at least $1/2$ for $M$ sufficiently
large, it follows that $\Delta_{k}^{(j)}$ remains subgaussian conditionally on
$\mathscr{F}_{k_{r-1}},$ with subgaussian constant only an absolute constant
more than the unconditioned standard deviation of $\Delta_{k}^{(j)}$ given
$\mathscr{F}_{k_{r-1}}.$ As $\Delta_{k}^{(j)}$ remains centered under
$\mathbb{P}_{M},$ using standard manipulations (see for example [Ver18,
Proposition 2.5.2]) it follows that there is another absolute constant $C>0$
so that for all $\mu\in{\mathbb{R}}$

$\mathbb{E}_{M}\left[e^{\mu\Delta_{k_{r}}^{(j)}}\middle|\mathscr{F}_{k_{r-1}}\right]\leq\exp\left(\mu\Delta_{k_{r-1}}^{(j)}+\frac{\mu^{2}C\varkappa|\Delta^{(j)}_{k_{r-1}}|^{2}}{\beta
k_{r-1}}+\frac{\mu^{2}C\varkappa}{\beta\omega k_{r-1}}\right).$

Likewise, the desired concentration inequality follows for the maximum. ∎

######  Lemma 5.4.

Suppose that $j\in\mathcal{D}_{n/k_{1}}$ and that $\theta\in{\mathbb{R}}$
satisfies

$|e(\tfrac{\theta_{j}}{2\pi})-e(\tfrac{\theta}{2\pi})|\leq\frac{\omega}{2\varkappa}.$

For all $\delta$ sufficiently small and all $C>0$, if $n_{1}^{+}/\varkappa\geq
n^{\delta}$ and $\omega\geq n^{\delta}$ then for all $n$ sufficiently large

$\mathbb{P}_{M}\left[\max_{{n_{1}^{+}}\leq k\leq
n}|R_{k}^{(j)}(\theta)+i\Delta_{k}^{(j)}(\theta)|\geq
n^{\delta}/\sqrt{\omega}\right]\leq n^{-C}.$

###### Proof.

Using Lemma 5.3, for any $t\geq 0$

$\mathbb{P}_{M}\left[\max_{k_{r-1}\leq
k<k_{r}}|\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)}|\geq
t\bigl{(}\sqrt{\omega}|\Delta_{k_{r-1}}^{(j)}|+1\bigr{)}\middle|\mathscr{F}_{k_{r-1}}\right]\leq
2\exp\left(-\frac{t^{2}k_{r-1}\omega\beta}{C\varkappa}\right).$