$\left\|P^{n}f\right\|_{2}^{2}\leqslant\gamma\left(n\right)\cdot\left\|f\right\|_{{\rm
osc}}^{2}$, where $\gamma\left(n\right)\to 0$. It is important to note that
$P$ (rather than $P^{*}P$) possessing a WPI does not necessarily imply that
$\left\|P^{n}f\right\|_{2}\to 0$ for bounded $f$. Indeed, a reversible,
periodic Markov kernel may satisfy a WPI, yet $\left\|P^{n}f\right\|_{2}$
cannot converge to $0$ for all bounded functions. Thus the goal of this
subsection is to give conditions which allow one to connect a WPI for $P$ with
convergence of $P$.

######  Remark 59.

When $P$ is reversible, it is possible to deduce the existence of a WPI for
$P^{2}=P^{*}P$ from a WPI for $P$, provided that one has some additional
control on the left spectral gap; see [1, Section 2.2.1]. In turn, the
existence of a WPI for $P$ can often be straightforwardly deduced from
Corollary 51 by establishing irreducibility of $P$.

######  Proposition 60.

If $P$ is convergent, then $S$ is convergent.

###### Proof.

If $P$ is convergent, then Lemma 56 implies that $P^{2}$ is RUPI and satisfies
a WPI. We may then deduce that $S^{2}$ satisfies a WPI because for any
$f\in{\rm L}_{0}^{2}\left(\mu\right)$,

$\displaystyle\mathcal{E}\left(S^{2},f\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\cdot\left\\{\mathcal{E}\left(P^{2},f\right)+\mathcal{E}\left(\left(P^{*}\right)^{2},f\right)+\mathcal{E}\left(PP^{*},f\right)+\mathcal{E}\left(P^{*}P,f\right)\right\\}$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{4}\cdot\mathcal{E}\left(P^{2},f\right).$

It follows that $S$ is convergent by Theorem 8. ∎

The converse does not hold in general, as seen by the following example.

######  Example 61 (Walks on the circle).

For $x,y\in\mathsf{E}=\left\\{1,\ldots,m\right\\}$ for $m$ odd, and let
$P\left(x,y\right)={\bf
1}_{\left\\{1,\ldots,m-1\right\\}}\left(x\right)\cdot{\bf
1}_{\left\\{x+1\right\\}}\left(y\right)+{\bf
1}_{\left\\{m\right\\}}\left(x\right)\cdot{\bf
1}_{\left\\{1\right\\}}\left(y\right)$ so that $P^{*}\left(x,y\right)={\bf
1}_{\left\\{2,\ldots,m\right\\}}\left(x\right)\cdot{\bf
1}_{\left\\{x-1\right\\}}\left(y\right)+{\bf
1}_{\left\\{1\right\\}}\left(x\right)\cdot{\bf
1}_{\left\\{m\right\\}}\left(y\right)$. Then the Markov chain associated with
$P$ is deterministic and one can deduce that $P$ is not convergent. On the
other hand, $S=\left(P+P^{*}\right)/2$ encodes a random walk on
$\left\\{1,\ldots,m\right\\}$ and is convergent.

######  Lemma 62.

For a $\mu$-invariant kernel $P$,
$P\left(x,\left\\{x\right\\}\right)=P^{*}\left(x,\left\\{x\right\\}\right)$
for $\mu$-almost all $x$.

###### Proof.

Let $D=\left\\{\left(x,y\right)\in\mathsf{E}^{2}:x=y\right\\}$,
$s\left(x\right):=P\left(x,\left\\{x\right\\}\right)$ and
$s^{*}\left(x\right):=P^{*}\left(x,\left\\{x\right\\}\right)$ for
$x\in\mathsf{E}$. For any $B\in\mathcal{E}$, we have

$\mu\left({\bf 1}_{B}\cdot s\right)=\mu\otimes P\left(D\cap\left(B\times
B\right)\right)=\mu\otimes P^{*}\left(D\cap\left(B\times
B\right)\right)=\mu\left({\bf 1}_{B}\cdot s^{*}\right),$

and so taking in turn
$B=\left\\{x\in\mathsf{E}:s\left(x\right)>s^{*}\left(x\right)\right\\}$ and
$B=\left\\{x\in\mathsf{E}:s\left(x\right)<s^{*}\left(x\right)\right\\}$, we
deduce that

$\mu\left(\left(s-s^{*}\right)^{+}\right)=0=\mu\left(\left(s-s^{*}\right)^{-}\right),$

and hence that $s=s^{*}$ $\mu$-almost everywhere. ∎

###### Proof of Theorem 47.

(b. $\Rightarrow$ c.) follows from Theorem 8, and (c. $\Rightarrow$ a.)
follows from Lemma 56. We now show (a. $\Rightarrow$ b.). We apply Lemma 57,
and hence by chaining WPIs [1, Theorem 33], we conclude that $P^{*}P$
satisfies an WPI.

We now show that the cycle (a. $\Rightarrow$ d. $\Rightarrow$ e. $\Rightarrow$
a.) can also be deduced. Observe that
$P^{*}\left(x,\left\\{x\right\\}\right)=P\left(x,\left\\{x\right\\}\right)$ by
Lemma 62, and $P$ satisfying a WPI is equivalent to $P^{*}$ satisfying a WPI,
since $\mathcal{E}\left(P,f\right)=\mathcal{E}\left(P^{*},f\right)$. Because
$\left(P^{*}\right)^{*}P^{*}=PP^{*}$, we have that (a. $\Rightarrow$ d.) is
equivalent to (a. $\Rightarrow$ b.) and (d. $\Rightarrow$ e.) is equivalent to
(b. $\Rightarrow$ c.) and (e. $\Rightarrow$ a.) is equivalent to (c.
$\Rightarrow$ a.). ∎

####  3.2.3 WPIs and convergence in general

######  Proposition 63.

$P$ satisfies a WPI if and only if for some (and hence any)
$\epsilon\in\left(0,1\right)$, $P_{\epsilon}^{*}P_{\epsilon}$ satisfies a WPI.

###### Proof.

($\Rightarrow)$ If $P$ satisfies a WPI then one may deduce that $P_{\epsilon}$
satisfies an WPI since
$\mathcal{E}\left(P_{\epsilon},f\right)=\left(1-\epsilon\right)\cdot\mathcal{E}\left(P,f\right)$.
Since $P_{\epsilon}$ satisfies (PH), Theorem 47 implies that
$P_{\epsilon}^{*}P_{\epsilon}$ satisfies a WPI. ($\Leftarrow)$ If
$P_{\epsilon}^{*}P_{\epsilon}$ satisfies a WPI then similarly Theorem 47
implies that $P_{\epsilon}$ satisfies a WPI and so $P$ satisfies a WPI. ∎

######  Proposition 64.

If $S$ (or $P$) is convergent then $P_{\epsilon}^{*}P_{\epsilon}$ satisfies a
WPI, and thus $P_{\epsilon}$ is convergent.

###### Proof.

If $S$ is convergent then $S$ satisfies a WPI by Lemma 56. Hence $P$ satisfies
a WPI by Remark 6 and we conclude by Proposition 63. ∎

Our counterexample of a Markov chain that is convergent but for which
$\left(P^{*}\right)^{k}P^{k}$ does not admit a WPI for any $k\in\mathbb{N}$ is
the following.

######  Example 65.

Let $\mathsf{E}=\left\\{1,2,\ldots\right\\}^{2}$, and $\nu$ a probability mass
function on $\left\\{1,2,\ldots\right\\}$ such that
$\nu\left(1\right)\in\left(0,1\right)$ and $\nu$ has a finite mean. Define

$P\left(i,j;i^{{}^{\prime}},j^{{}^{\prime}}\right)=\begin{cases}1&j<i,i^{\prime}=i,j^{\prime}=j+1,\\\
\nu\left(i^{\prime}\right)&j=i,j^{\prime}=1,\\\
1&\nu\left(i\right)\cdot\mathbf{1}\left\\{j\leqslant
i\right\\}=0,\left(i^{\prime},j^{\prime}\right)=\left(1,1\right),\\\
0&\text{otherwise}.\end{cases}$

The intuition is that the Markov chain moves to the right along “level” $i$
deterministically until it reaches the point $\left(i,i\right)$, at which
point it jumps to the start of another level $\left(K,1\right)$ where
$K\sim\nu$. The third statement is concerned with initialization of the chain
outside the support of the invariant distribution $\mu$, given below.

######  Proposition 66.

The Markov kernel $P$ in Example 65

  1. a).

has invariant distribution

$\mu\left(i,j\right)=\frac{\nu\left(i\right)\cdot\mathbf{1}\left\\{j\leqslant
i\right\\}}{\sum_{k=1}^{\infty}\nu\left(k\right)\cdot k};$

  2. b).

is $\mu$-irreducible with an accessible, aperiodic atom $\left(1,1\right)$ and
its Markov chain converges to $\mu$ in total variation from any starting point
.

  3. c).

its Markov chain is convergent,

  4. d).

is such that when $\nu\left(i\right)>0$ for all $i\geqslant 1$,

    1. i

then $\left(P^{*}\right)^{k}P^{k}$ is reducible for any $k\geqslant 1$. More
specifically for any $1\leqslant k<i$

$\left(P^{*}\right)^{k}P^{k}\left(i,i;i,i\right)=1;$

    2. ii

$\left(P^{*}\right)^{k}P^{k}$ does not admit a WPI for any $k\in\mathbb{N}$.

###### Proof.

The first two statements can be checked directly and using e.g., [18, Theorem
7.6.4]. The third statement follows from (17). Now, by viewing $P^{*}$ as the
time-reversal of $P$, and satisfying $\mu\left(i,j\right)\cdot
P^{*}\left(i,j;i^{{}^{\prime}},j^{{}^{\prime}}\right)=\mu\left(i^{{}^{\prime}},j^{{}^{\prime}}\right)\cdot
P\left(i^{{}^{\prime}},j^{{}^{\prime}};i,j\right)$, we may define

$P^{*}\left(i,j;i^{{}^{\prime}},j^{{}^{\prime}}\right)=\begin{cases}1&j>1,i^{\prime}=i,j^{\prime}=j-1,\\\
\nu\left(i^{{}^{\prime}}\right)&j=1,j^{\prime}=i^{\prime},\\\
1&\nu\left(i\right)\cdot\mathbf{1}\left[j\leqslant
i\right]=0,\big{(}i^{{}^{\prime}},j^{{}^{\prime}}\big{)}=\left(1,1\right),\\\
0&\text{otherwise}.\end{cases}$

Consider first the case where $i_{0}\in\left\\{2,3,\ldots\right\\}$,
$\nu\left(i\right)>0$ for all $i\leqslant i_{0}$ and $\nu\left(i\right)=0$ for
$i>i_{0}$. This means there is a maximum level length of $i_{0}$. We see that
if $k<i_{0}$ then

$\left(P^{*}\right)^{k}P^{k}\left(i_{0},i_{0};i_{0},i_{0}\right)=1,$

since $\left(P^{*}\right)^{k}\left(i_{0},i_{0};i_{0}-k,i_{0}-k\right)=1$ and
$P^{k}\left(i_{0}-k,i_{0}-k;i_{0},i_{0}\right)=1$. Hence
$\left(P^{*}\right)^{k}P^{k}$ is reducible for any $k<i_{0}-1$. On the other
hand, for $k\geqslant i_{0}$, we may deduce that $\left(P^{*}\right)^{k}P^{k}$
is $\mu$-irreducible. In particular, since
$P^{*}\left(1,1;1,1\right)=P\left(1,1;1,1\right)=\nu\left(1\right)\in\left(0,1\right)$,
we see that $\left(P^{*}\right)^{k}\left(i,j;1,1\right)>0$ for all
$\left(i,j\right)\in\mathsf{E}$, from which one may deduce that
$\left(P^{*}\right)^{k}P^{k}\left(i,j;i^{{}^{\prime}},j^{{}^{\prime}}\right)>0$
for all $i,j,i^{\prime},j^{\prime}\in\mathsf{E}$ such that
$\mu\left(i^{{}^{\prime}},j^{{}^{\prime}}\right)>0$. Note that since
$P\left(i,j;1,1\right)=1$ for all $\left(i,j\right)$ such that
$\mu\left(i,j\right)=0$, this is essentially a finite state space Markov chain
after 1 step, and hence convergence is geometric. Assume now that
$\nu\left(i\right)>0$ for all $i\in\left\\{1,2,\ldots\right\\}$. From above,
for any $k\in\mathbb{N}$ we may consider level $i>k$ and we see that
$\left(P^{*}\right)^{k}P^{k}\left(i,i;i,i\right)=1$ so
$\left(P^{*}\right)^{k}P^{k}$ is reducible. Hence, there does not exist
$k\in\mathbb{N}$ such that $\left(P^{*}\right)^{k}P^{k}$ is $\mu$-irreducible.

For the last statement, let $k\in\mathbb{N}$ be arbitrary. Let
$A_{k}=\left\\{\left(i,i\right):i>k\right\\}$ and $f_{k}={\bf
1}_{A_{k}}-\mu\left(A_{k}\right)$, which satisfies $\mu\left(f_{k}\right)=0$.
Then
$\left\|P^{k}f_{k}\right\|_{2}^{2}=\left\langle\left(P^{*}\right)^{k}P^{k}f_{k},f_{k}\right\rangle=\left\langle
f_{k},f_{k}\right\rangle=\left\|f_{k}\right\|_{2}^{2}$, from d)i, so
$\mathcal{E}\big{(}(P^{*})^{k}P^{k},f_{k}\big{)}=0$. Since
$\left\|f_{k}\right\|_{2}>0$, $\left(P^{*}\right)^{k}P^{k}$ cannot satisfy a
WPI. ∎

######  Remark 67.

One may assume that the non-existence of a WPI in Example 65 is the result of
slow convergence for $P$. This is not the case, as it can be shown that for
the choice $\nu\left(i\right)=\left(1-a\right)\cdot a^{i-1}$ for some
$a\in\left(0,1\right)$, the resulting Markov chain is
$\left(\left\|\cdot\right\|_{\mathrm{osc}}^{2},\left\\{C\cdot\rho^{n}\rho^{n}\colon
C>1,\rho\in\left[0,1\right),n\in\mathbb{N}\right\\}\right)$-convergent [3,
Proposition 67]. We note that this scenario differs from what is normally
understood as hypocoercivity, for which one may hope to establish
$\big{(}\left\|\cdot\right\|_{2}^{2},\left\\{C\cdot\rho^{n}\rho^{n}\colon
C>1,\rho\in\left[0,1\right),n\in\mathbb{N}\right\\}\big{)}$-convergence i.e.
for all $f\in\mathrm{L}^{2}\left(\mu\right)$

$\left\|P^{n}f\right\|_{2}^{2}\leqslant
C\cdot\rho^{n}\cdot\left\|f\right\|_{2}^{2}\,,$

despite the degeneracy of the our generator ${\rm
Id}-\left(P^{*}\right)^{k}P^{k}$ for any $k\in\mathbb{N}$. This would however
imply the existence of, for example, a SPI
$\mathcal{E}\left(\left(P^{*}\right)^{k}P^{k},f\right)\geqslant\frac{1}{2}\left\|f\right\|_{2}^{2}$
for some $k\in\mathbb{N}$, therefore contradicting Proposition 66.
Hypocoercivity in the sense above indeed holds whenever $\nu$ has finite
support.

###  3.3 WPIs from Close Coupling and Isoperimetry

This section is concerned with establishing quantitative bounds on WPIs and
mixing times for Markov chains. Building on Section 3.1, the goal is to
control the weak conductance profile of the Markov kernel using isoperimetric
properties of the target distribution and a close coupling condition of the
Markov kernel.

####  3.3.1 Main results

We first recall some concepts and results from [2].

######  Assumption 68.

The probability distribution $\pi$ has a positive density with respect to
Lebesgue measure on $\mathsf{E}=\mathbb{R}^{d}$, given by
$\pi\propto\exp\left(-U\right)$, for some potential
$U:\mathbb{R}^{d}\to\mathbb{R}$.

######  Definition 69 (Three-set isoperimetric inequality).

A probability measure $\pi$ satisfies a _three-set isoperimetric inequality_
with metric $\mathsf{d}$ and function
$F:\left(0,\frac{1}{2}\right]\to\left[0,\infty\right)$ if for all measurable
partitions of the state space $\mathsf{E}=S_{1}\sqcup S_{2}\sqcup S_{3}$,

$\pi\left(S_{3}\right)\geqslant\mathsf{d}\left(S_{1},S_{2}\right)\cdot
F\left(\min\left\\{\pi\left(S_{1}\right),\pi\left(S_{2}\right)\right\\}\right).$

######  Definition 70 (Isoperimetric Profile).

For $A\in\mathscr{E}$ and $r\geqslant 0$, let
$A_{r}:=\left\\{x\in\mathsf{E}:\mathsf{d}\left(x,A\right)\leqslant
r\right\\}$, and define the Minkowski content of $A$ under $\pi$ with respect
to $\mathsf{d}$ by

$\pi^{+}\left(A\right)=\lim\inf_{r\to
0^{+}}\frac{\pi\left(A_{r}\right)-\pi\left(A\right)}{r}.$

The isoperimetric profile of $\pi$ with respect to the metric $\mathsf{d}$ is

$I_{\pi}\left(p\right):=\inf\left\\{\pi^{+}\left(A\right):A\in\mathscr{E},\pi\left(A\right)=p\right\\},\qquad
p\in\left(0,1\right).$

We note briefly that the isoperimetric profile can be controlled explicitly in
many cases of interest; specific examples to this effect are provided in
Section 3.3.2.

######  Definition 71.

We say that $\tilde{I}_{\pi}:\left(0,1\right)\to\left(0,\infty\right)$ is an
isoperimetric minorant of $\pi$ if $\tilde{I}_{\pi}\leqslant I_{\pi}$
pointwise. We furthermore say that $\tilde{I}_{\pi}$ is regular if it is
symmetric about $\frac{1}{2}$, continuous, and monotone increasing on
$\left(0,\frac{1}{2}\right]$.

To begin with, we recall that given any regular isoperimetric minorant of our
target measure, one can directly deduce a three-set isoperimetric inequality
for the same measure.

######  Lemma 72.

Let $\pi$ have a regular isoperimetric minorant $\tilde{I}_{\pi}$ w.r.t. the
metric $\mathsf{d}$. Then $\pi$ satisfies a three-set isoperimetric inequality
with metric $\mathsf{d}$ and function $F=\tilde{I}_{\pi}$ on
$\left(0,\frac{1}{2}\right]$.

######  Definition 73.

For a metric $\mathsf{d}$ on $\mathsf{E}$ and $\epsilon,\delta>0$, a Markov
kernel $P$ evolving on $\mathsf{E}$ is
$\left(\mathsf{d},\delta,\varepsilon\right)$-close coupling if

$\mathsf{d}\left(x,y\right)\;\leqslant\delta\Rightarrow\left\|P\left(x,\cdot\right)-P\left(y,\cdot\right)\right\|_{{\rm
TV}}\leqslant 1-\varepsilon,\qquad x,y\in\mathsf{E}.$

######  Lemma 74.

Suppose that $\pi$ satisfies a three-set isoperimetric inequality with metric
$\mathsf{d}$ and function $F$ monotone increasing on
$\left[0,\frac{1}{2}\right)$. Let $P$ be a
$\left(\mathsf{d},\delta,\varepsilon\right)$-close coupling, $\pi$-invariant
Markov kernel. Then for any $A\in\mathscr{E}$ with
$\pi\left(A\right)\leqslant\frac{1}{2}$,

$\frac{\left(\pi\otimes P\right)\left(A\times
A^{\complement}\right)}{\pi\left(A\right)}\geqslant\sup_{\theta\in\left[0,1\right]}\min\left\\{\frac{1}{2}\cdot\left(1-\theta\right)\cdot\varepsilon,\frac{1}{4}\cdot\varepsilon\cdot\delta\cdot\theta\cdot\left(F/{\rm
id}\right)\left(\theta\cdot\pi\left(A\right)\right)\right\\}.$

In contrast to our earlier work, in the heavy-tailed case, isoperimetric
profiles and minorants are typically convex, meaning that
$I_{\pi}/\mathrm{id}$ is usually increasing and vanishes at $0$. As such,
taking an infimum over sufficiently small sets $A$ in the above will give
trivial bounds. We thus instead take infima over sufficiently large sets $A$.

######  Corollary 75.

Suppose that $\tilde{I}_{\pi}$ is a regular, convex isoperimetric minorant of
$\pi$ w.r.t. the metric $\mathsf{d}$. Let $P$ be a
$\left(\mathsf{d},\delta,\varepsilon\right)$-close coupling, $\pi$-invariant
Markov kernel. Then for any $v\in\left(0,\frac{1}{2}\right]$,

$\displaystyle\Phi_{P}^{\mathrm{W}}\left(v\right)$
$\displaystyle\geqslant\sup_{\theta\in\left[0,1\right]}\min\left\\{\frac{1}{2}\cdot\left(1-\theta\right)\cdot\varepsilon,\frac{1}{4}\cdot\varepsilon\cdot\delta\cdot\theta\cdot\left(\tilde{I}_{\pi}/{\rm
id}\right)\left(\theta\cdot v\right)\right\\}$
$\displaystyle\geqslant\frac{1}{4}\cdot\varepsilon\cdot\min\left\\{1,\frac{1}{2}\cdot\delta\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot v}\right\\}.$

######  Theorem 76.

Let $\tilde{I}_{\pi}$ be a regular, convex isoperimetric minorant of $\pi$
w.r.t. the metric $\mathsf{d}$. Let $P$ be a positive, $\pi$-reversible and
$\left(\mathsf{d},\delta,\varepsilon\right)$-close coupling Markov kernel.
Assume $2\cdot\delta\cdot\tilde{I}_{\pi}\left(\frac{1}{4}\right)\leqslant 1$.
Then for

$\sup\big{\\{}\left\|P^{n}f-\pi\left(f\right)\right\|_{2}^{2}\colon\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant
1\big{\\}}\leqslant\varepsilon_{{\rm Mix}}\quad\text{for some
}\varepsilon_{{\rm Mix}}\in\left(0,\frac{1}{4}\right),$

to hold, it suffices to take

$\displaystyle n$ $\displaystyle\geqslant
1+2^{8}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\int_{2^{-3}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}.$

Let $\nu\ll\pi$ be a probability measure with
$u:=\left\|\frac{\mathrm{d}\nu}{\mathrm{d}\pi}\right\|_{\mathrm{osc}}^{2}<\infty$.
Then in order to ensure that

$\chi^{2}\left(\nu P^{n},\pi\right)\leqslant\varepsilon_{{\rm
Mix}}\quad\text{for some }\varepsilon_{{\rm Mix}}>0,$

it suffices that

$n\geqslant
1+2^{8}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\int_{2^{-3}\cdot\varepsilon_{{\rm
Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}.$

###### Proof.

For any $f\in\mathrm{L}^{2}\left(\mu\right)$ and $n\in\mathbb{N}$, set
$v_{n}:=\left\|P^{n}f-\pi\left(f\right)\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant
1/4$. Then from Corollaries 36 and 75 we have for any $n\in\mathbb{N}$,

$\displaystyle n$ $\displaystyle\leqslant 2^{2}\cdot\int_{2^{-2}\cdot
v_{n}}^{2^{-4}}\frac{\mathrm{d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(v\right)^{2}}$
$\displaystyle\leqslant 2^{2}\cdot\int_{2^{-2}\cdot
v_{n}}^{2^{-4}}2^{6}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\frac{\left(\frac{1}{2}\cdot
v\right)^{2}}{\left(\frac{1}{2}\cdot
v\right)\cdot\left[\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)\right]^{2}}\mathrm{d}\left(\frac{1}{2}\cdot v\right)$
$\displaystyle\leqslant
2^{8}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\int_{2^{-3}\cdot
v_{n}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}.$

Therefore for $n\in\mathbb{N}$ such that

$n\geqslant
1+2^{8}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\int_{2^{-3}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}},$

we must have
$\sup\big{\\{}\left\|P^{n}f-\pi\left(f\right)\right\|_{2}^{2}\colon\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant
1\big{\\}}\leqslant\varepsilon_{{\rm Mix}}$. The second result follows by
considering $f=\frac{\mathrm{d}\nu}{\mathrm{d}\pi}$ and noting that
$\Psi\left(f\right)=u$. ∎

####  3.3.2 Examples of (Convex) Isoperimetric Profiles

In this section, we give some examples of heavy-tailed probability measures
for which we can give bounds on the isoperimetric profile. This will be based
around results for some illustrative model classes in general dimension with
some degree of convexity, as well as some results on how profiles can behave
under tensorisation. Our main generic examples are borrowed from [14], and
some more specific cases follow from analyses in [9, 14]. The general theme is
that for quasi-convex potentials, one can still often obtain reasonable
isoperimetric estimates, at least in terms of capturing the correct asymptotic
behaviour with respect to the parameter $p$. In contrast, dependence on
dimension (in the non-tensorised case) and constant pre-factors can be a bit
looser. As in the log-concave case, transfer principles (e.g. Lipschitz
transport, bounded change-of-measure) remain relevant and straightforward to
apply.

When $\mathsf{E}=\mathbb{R}$, consider probability distributions of the form

$\pi\left(\mathrm{d}x\right)\propto\exp\left(-U\left(\left|x\right|\right)\right)\mathrm{d}x\,,$

with $U\colon\mathbb{R}_{+}\rightarrow\mathbb{R}$ concave. Letting $\pi$ also
denote the probability density and $G_{\pi}$ the corresponding cumulative
distribution function, define $J_{\pi}:=\pi\circ G_{\pi}^{-1}$. Then [12,
Corollary 13.10], [14, Eq. (5.20)] establish that

$\displaystyle I_{\pi}\left(p\right)$
$\displaystyle=\min\left\\{J_{\pi}\left(p\right),2\cdot
J_{\pi}\left(\frac{1}{2}\cdot p\right)\right\\},\quad
p\in\left[0,\frac{1}{2}\right],$

corresponding to the conclusion that the extremal sets in the isoperimetric
problem take the form of either semi-infinite intervals, symmetric pairs of
semi-infinite intervals, or complements thereof. [14, Proof of Corollary 5.16]
later establish that for $d\geqslant 1$, the product measures $\pi^{\otimes
d}$ on $\mathsf{E}=\mathbb{R}^{d}$ for $d\geqslant 1$ have isoperimetric
profiles such that for $p\in\left[0,\frac{1}{2}\right]$,

$d\cdot\frac{c_{1}}{c_{2}}\cdot J_{\pi}\left(\frac{1}{2\cdot
c_{1}}\cdot\frac{p}{d}\right)\leqslant I_{\pi^{\otimes
d}}\left(p\right)\leqslant d\cdot\frac{c_{1}}{c_{2}}\cdot
J_{\pi}\left(\frac{1}{c_{1}}\cdot\frac{p}{d}\right)$

with $c_{1}=2\cdot\sqrt{6}$ and $c_{2}=2\cdot\left(1+c_{1}\right)$. Convenient
and accurate bounds for $J_{\pi}$ can be found with mild additional conditions
on $U$.

######  Proposition 77 ([14, Corollary 5.23]).

Assume $U\colon\mathbb{R}_{+}\rightarrow\mathbb{R}$ to be a non-decreasing
concave function such that

  1. a).

$\lim_{x\rightarrow\infty}\frac{U\left(x\right)}{x}=0$ and
$U\left(0\right)<\log 2$;

  2. b).

U is $\mathrm{C}^{2}$ in a neighbourhood of $\infty$;

  3. c).

$U^{\theta}$ for some $\theta>1$ is convex.

Then,

  1. a).

there exist $k_{1},k_{2}>0$ such that for $p\in\left[0,1\right]$

$k_{1}\cdot p\cdot\left(U^{{}^{\prime}}\circ
U^{-1}\right)\left(\log\left(1/p\right)\right)\leqslant
J_{\pi}\left(p\right)\leqslant k_{2}\cdot p\cdot\left(U^{{}^{\prime}}\circ
U^{-1}\right)\left(\log\left(1/p\right)\right);$

  2. b).

in particular, there exists $c>0$ such that for any $d\geqslant 1$

$I_{\pi^{\otimes d}}\left(p\right)\geqslant c\cdot
p\cdot\left(U^{{}^{\prime}}\circ
U^{-1}\right)\left(\log\left(\frac{d}{p}\right)\right)\quad
p\in\left[0,\frac{1}{2}\right].$

Together, these results suggest that this estimate of the isoperimetric
profile of $\pi^{\otimes d}$ is accurate up to a factor $2$ in the dimension
$d$. In the examples below, we provide lower bounds for $I_{\pi^{\otimes
d}}\left(p\right)$, but explicit upper bounds can also be found in [14, p.
376], establishing correct dependence on dimension for the two first examples.

######  Example 78.

For some concrete examples, this approach yields that for
$p\in\left(0,\frac{1}{2}\right)$,

  * •

If $U\left(x\right)=\left|x\right|^{\eta}$ with $\eta\in\left(0,1\right)$,
then there exists $c>0$ (dependent on $\eta$ only) such that $I_{\pi^{\otimes
d}}\left(p\right)\geqslant c\cdot
p\cdot\left(\log\left(\frac{d}{p}\right)\right)^{-\left(1/\eta-1\right)}$;
[14, Proposition 5.25; Remark 5.26].

  * •

If
$U\left(x\right)=\left(1+\eta\right)\cdot\log\left(1+\left|x\right|\right)$,
then there exists $c$ dependent on $\eta$ only such that $I_{\pi^{\otimes
d}}\left(p\right)\geqslant c\cdot d^{-1/\eta}\cdot p^{1+1/\eta}$ [14, Remark
5.28]; note that this is not implied directly by [14, Corollary 5.23] but
follows from a separate result.

  * •

If
$U\left(x\right)=\left|x\right|^{\eta_{1}}\cdot\left(\log\left(\eta_{3}+\left|x\right|\right)\right)^{\eta_{2}}$,
$\eta_{1}\in\left(0,1\right)$, $\eta_{2}\in\mathbb{R}$ and
$\eta_{3}=\exp\left(\frac{2|\eta_{2}|}{\eta_{1}\cdot\left(1-\eta_{1}\right)}\right)$
(so that $U$ is globally concave on $\mathbb{R}$), then there exists $c>0$
(dependent on $\eta_{1},\eta_{2},\eta_{3}$ only) such that

$I_{\pi^{\otimes d}}\left(p\right)\geqslant c\cdot
p\cdot\left(\log\left(\frac{d}{p}\right)\right)^{-\left(1/\eta_{1}-1\right)}\cdot\left(\log\left(\log\left(e+\frac{d}{p}\right)\right)\right)^{\eta_{2}/\eta_{1}},$

(see [14, Below Remark 5.26]).

In particular, for sub-exponential-type targets, the effect of tensorisation
is roughly to attenuate the isoperimetric profile by a factor of $\left(\log
d\right)^{-\left(1/\eta-1\right)}$. In contrast, for targets with polynomial
tails, the profile is attenuated more dramatically by a polynomial factor of
$d^{-1/\alpha}$.

From Theorem 76 this leads to the following mixing time upper bounds

  * •

For $I_{\pi^{\otimes d}}\left(p\right)\geqslant c\cdot d^{-1/\eta}\cdot
p^{1+1/\eta}$ we have

$\displaystyle\int_{2^{-4}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-6}}\frac{v\,\mathrm{d}v}{I_{\pi^{\otimes d}}\left(v\right){}^{2}}$
$\displaystyle\leqslant c^{-2}\cdot
d^{2/\eta}\cdot\int_{2^{-4}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-6}}v^{-1-2/\eta}{\rm d}v$ $\displaystyle\leqslant\eta\cdot
2^{8/\eta-1}\cdot c^{-2}\cdot d^{2/\eta}\cdot\varepsilon_{{\rm
Mix}}^{-2/\eta},$

that is, it is sufficient to take

$\displaystyle n$ $\displaystyle\geqslant 1+c^{-2}\cdot\eta\cdot
2^{8/\eta+8}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot
d^{2/\eta}\cdot\varepsilon_{{\rm Mix}}^{-2/\eta}$
$\displaystyle=\Omega\left(\varepsilon^{-2}\cdot\delta^{-2}\cdot
d^{2/\eta}\cdot\varepsilon_{{\rm Mix}}^{-2/\eta}\right),$

in Theorem 76.

  * •

For $I_{\pi^{\otimes d}}\left(p\right)\geqslant c\cdot
p\cdot\log\left(\frac{d}{p}\right)^{1-1/\eta}$ , $\eta\in\left(0,1\right)$ we
have

$\displaystyle\int_{2^{-4}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-6}}\frac{v\,\mathrm{d}v}{I_{\pi^{\otimes d}}\left(v\right)^{2}}$
$\displaystyle\leqslant c^{-2}\cdot\int_{2^{-4}\cdot\varepsilon_{{\rm
Mix}}}^{2^{-6}}\frac{{\rm d}v}{v\cdot\log\left(\frac{d}{v}\right)^{2-2/\eta}}$
$\displaystyle\leqslant
c^{-2}\cdot\frac{\eta}{2-\eta}\cdot\log\left(2^{4}\cdot
d\cdot\varepsilon_{{\rm Mix}}^{-1}\right)^{2/\eta-1},$

that is it is sufficient to take

$\displaystyle n$ $\displaystyle\geqslant 1+2^{9}\cdot
c^{-2}\cdot\frac{\eta}{2-\eta}\cdot\varepsilon^{-2}\cdot\delta^{-2}\cdot\left(\log\left(2^{4}\cdot
d\cdot\varepsilon_{{\rm Mix}}^{-1}\right)\right)^{2/\eta-1}$
$\displaystyle=\Omega\left(\varepsilon^{-2}\cdot\delta^{-2}\cdot\left(\log\left(d\cdot\varepsilon_{{\rm
Mix}}^{-1}\right)\right)^{2/\eta-1}\right),$

in Theorem 76.

Some attempts have been made in [14] to extend the class of distributions for
which lower bounds on the isoperimetric profile can be found. In what follows,
let $V\colon\mathsf{E}=\mathbb{R}^{d}\rightarrow\mathbb{R}$ denote a generic
coercive convex function satisfying a growth condition of the form

$\left|x\right|\geqslant R\implies V\left(x\right)\geqslant
V\left(0\right)+\delta\cdot\left|x\right|,$

for some $R,\delta>0$. Here, $V$ will dictate the shape of the contours of our
target, and our potential $U$ will be obtained as a monotone, sublinear
function of $V$.

######  Example 79.

[14, Proposition 4.3] For Cauchy-type targets of the form
$\pi\left(\mathrm{d}x\right)\propto
V\left(x\right)^{-\left(d+\eta\right)}\,\mathrm{d}x$ with $\eta>0$, it holds
that $I_{\pi}\left(p\right)\geqslant c\cdot p^{1+1/\eta}$ for
$p\in\left[0,\frac{1}{2}\right]$, for some constant $c$ which depends on
$\left(V,\eta,d\right)$. A general result of [9, Theorem 1.2] shows that one
can in fact take $c$ to depend only on $\eta$ and
$\mathrm{med}_{\pi}\left(\left|x\right|\right)$.

As an application of Bobkov’s result, one can see that the multivariate
Student-T distribution
$\pi\left(\mathrm{d}x\right)\propto\left(\tau+\left|x\right|_{2}^{2}\right)^{-\frac{d+\tau}{2}}\,\mathrm{d}x$
admits an isoperimetric minorant of the form $I_{\pi}\left(p\right)\geqslant
c\cdot d^{-1/2}\cdot p^{1+1/\tau}$, where $c$ can be bounded uniformly from
below for $\min\left\\{d,\tau\right\\}\in\Omega\left(1\right)$, see Lemma 95.
Related functional inequalities suggest that the dimensional factor $d^{-1/2}$
is likely to be extraneous (see e.g. the (adimensional) weighted inequalities
of [11]), but we are not aware of a formal proof of this.

######  Example 80.

[14, Proposition 4.5] For sub-exponential-type targets of the form
$\pi\left(\mathrm{d}x\right)\propto\exp\left(-V\left(x\right)^{\eta}\right)\,\mathrm{d}x$
with $\eta\in\left(0,1\right)$, it holds that $I_{\pi}\left(p\right)\geqslant
c\cdot p\cdot\log\left(\frac{1}{p}\right)^{-\left(1/\eta-1\right)}$ for
$p\in\left[0,\frac{1}{2}\right]$, for some constant $c$ which depends on
$\left(V,\eta,d\right)$.

While these results allow one to characterize rates of convergence for fixed
$d$ for a very broad class of distributions, characterization of $c$, and
therefore mixing times, in terms of the dimension $d$ would require
considering particular cases, or even different techniques.

##  4 Application to Pseudo-marginal MCMC

Fix a probability distribution $\pi$ on a measure space $\mathsf{X}$, with a
density function with respect to some measure $\nu$ denoted $\varpi$. Pseudo-
marginal algorithms [4] extend the scope of the standard Metropolis–Hastings
algorithm to the scenario where the density $\varpi$ is intractable, but for
any $x\in\mathsf{X}$, nonnegative estimators $\hat{\varpi}\left(x\right)$ such
that
$\mathbb{E}\left[\hat{\varpi}\left(x\right)\right]=C\cdot\varpi\left(x\right)$
for some constant $C>0$ are available. The main idea of pseudo-marginal
algorithms is to replace the value of $\varpi\left(x\right)$ with
$\hat{\varpi}\left(x\right)$ in standard algorithms. The surprising fact is
that these algorithms are correct algorithms, in the sense that under standard
conditions they are guaranteed to produce samples from a distribution
arbitrarily close to $\pi$. This comes however at a price, as pseudo-
algorithms are slower to converge then their exact, or ‘marginal’,
counterparts which would use the intractable $\varpi(x)$. The degradation of
performance of the algorithm stemming from the presence of weights has been
investigated in [5, 6] and more recently in [1] and upper bounds on
convergence rates and/or complexity bounds were obtained. The object of this
section is to exploit “the left hand side Cheeger inequality” in Theorem 35 to
establish lower bounds on the rate of convergence of pseudo-marginal
algorithms.

Pseudo-marginal algorithms can be conveniently formulated as follows. Let
$\mu\left({\rm d}x,{\rm d}w\right)=\pi\left({\rm d}x\right)\cdot
Q_{x}\left({\rm d}w\right)\cdot w=\pi\left({\rm
d}x\right)\cdot\tilde{\pi}_{x}\left({\rm d}w\right)$ with
$\int_{\mathbb{R}_{+}}w\,Q_{x}\left({\rm d}w\right)=1$ on an extended space
$\mathsf{E}:=\mathsf{X\times\mathbb{R}_{+}}$. We will refer to these auxiliary
$w$ random variables as _weights_ or _perturbations_.

The _marginal (Metropolis–Hastings) algorithm_ , which uses the exact density
$\varpi={\rm d}\pi/{\rm d}\nu$ for some measure $\nu$ requires the
specification of a family of proposal distributions for
$\left\\{q\left(x,\cdot\right),x\in\mathsf{X}\right\\}$ (again having density
with respect to $\nu$) and has transition kernel

$\displaystyle P\left(x,{\rm d}y\right)$ $\displaystyle=\left[1\wedge
r\left(x,y\right)\right]\,q\left(x,{\rm d}y\right)+\delta_{x}\left({\rm
d}y\right)\cdot\rho\left(x\right),$ $\displaystyle\quad\text{where}\quad
r\left(x,y\right):=\frac{\varpi\left(y\right)\cdot
q\left(y,x\right)}{\varpi\left(x\right)\cdot q\left(x,y\right)},$ (22)

and $\rho$ is the rejection probability given by
$\rho\left(x\right):=1-\int\left[1\wedge
r\left(x,y\right)\right]\,q\left(x,{\rm d}y\right)$ for each $x\in\mathsf{X}$.
For brevity, we will also define the acceptance probability as
$a\left(x,y\right):=1\wedge r\left(x,y\right)$.

The pseudo-marginal Metropolis–Hastings kernel is given by

$\tilde{P}\left(x,w;{\rm d}y,{\rm
d}u\right)=\left[1\wedge\left\\{r\left(x,y\right)\cdot\frac{u}{w}\right\\}\right]\cdot
q\left(x,{\rm d}y\right)\cdot Q_{y}\left({\rm
d}u\right)+\delta_{x,w}\left({\rm d}y,{\rm
d}u\right)\cdot\tilde{\rho}\left(x,w\right),$ (23)

where the (joint) rejection probability $\tilde{\rho}\left(x,w\right)$ is
defined analogously. From [5, 6, 1] it is known that can upper bound the
convergence rate of the pseudo-marginal algorithm in terms of properties of
the marginal algorithm $P$ and tail properties of $\left\\{Q_{x}\left({\rm
d}w\right),x\in\mathsf{X}\right\\}$. The following establishes lower bounds.

######  Theorem 81.

Let $\tilde{P}$ be as in (23) with $q$ some $\nu$-reversible kernel. For any
$s>0$, let
$A_{s}:=\left\\{\left(x,w\right)\in\mathsf{X}\times\mathbb{R}_{+}\colon\varpi\left(x\right)w\geqslant
s\right\\}$, $\psi\colon\mathbb{R}_{+}\rightarrow\left[0,1\right]$ be such
that $\psi\left(s\right):=\mu\left(A_{s}\right)$ and for any
$v\in\left(0,1\right]$, consider the generalized inverse,

$\psi^{-}(v):=\sup\left\\{s>0:\mu\left(A_{s}\right)\geqslant v\right\\}.$ (24)

Then with $\Phi_{\tilde{P}}^{\mathrm{W}}$ the weak conductance of $\tilde{P}$
as given in (23), we have for $v>0$,

  1. a).

$\Phi_{\tilde{P}}^{\mathrm{W}}\left(v\right)\leqslant\frac{\int\pi\left(\mathrm{d}x\right)\tilde{\pi}_{x}\big{(}\varpi\left(x\right)W\geqslant\psi^{-}\left(v\right)\big{)}\int
q\left(x,\mathrm{d}y\right)\frac{\varpi\left(y\right)\tilde{\pi}_{y}\left(\varpi\left(y\right)W<\psi^{-}\left(v\right)\right)}{\psi^{-}\left(v\right)}}{\int\pi\left(\mathrm{d}x\right)\tilde{\pi}_{x}\big{(}\varpi\left(x\right)W\geqslant\psi^{-}\left(v\right)\big{)}};$

  2. b).

Assuming that
$\bar{\mathfrak{w}}:=\sup_{x\in\mathsf{X}}\varpi\left(x\right)<\infty$,

$\Phi_{\tilde{P}}^{\mathrm{W}}(v)\leqslant\frac{\bar{\mathfrak{w}}}{\psi^{-}\left(v\right)}.$

###### Proof.

The first statement follows from the fact that by definition for $0<v\leqslant
1/2$

$\Phi_{\tilde{P}}^{\mathrm{W}}\left(v\right)\leqslant\inf_{\left\\{s>0\colon
1/2\geqslant\mu\left(A_{s}\right)\geqslant
v\right\\}}\frac{\mathcal{E}\big{(}\tilde{P},\mathbf{1}_{A_{s}}\big{)}}{\mu\big{(}\mathbf{1}_{A_{s}}\big{)}},$

Lemma 84. The second statement is a direct consequence of the first.

∎

One could treat the case in which
$\sup_{x\in\mathsf{X}}\varpi\left(x\right)=\infty$, but as most practical
applications satisfy $\bar{\mathfrak{w}}<\infty$, we provide only the simpler
result here.

######  Example 82.

Assume that there exists a probability distribution $\tilde{\pi}_{*}$ and
$C,\underline{\mathfrak{w}}>0$ such that for any
$\left(x,A\right)\in\left\\{y\in\mathsf{X}\colon\varpi\left(y\right)\geqslant\underline{\mathfrak{w}}\right\\}\times\mathcal{B}\left(\mathbb{R}_{+}\right)$,
it holds that $\tilde{\pi}_{x}\left(A\right)\geqslant
C\cdot\tilde{\pi}_{*}\left(A\right)$. Then for any $s>0$, we can bound

$\displaystyle\mu\left(A_{s}\right)$ $\displaystyle\geqslant
C\cdot\int\mathbf{1}\left\\{\varpi\left(x\right)\geqslant\underline{\mathfrak{w}},w\geqslant
s/\varpi\left(x\right)\right\\}\cdot\pi\left({\rm
d}x\right)\cdot\tilde{\pi}_{*}\left({\rm d}w\right)$ $\displaystyle\geqslant
C\cdot\pi\left(\varpi\geqslant\underline{\mathfrak{w}}\right)\cdot\tilde{\pi}_{*}\left(W\geqslant
s/\underline{\mathfrak{w}}\right)$

and therefore

$\displaystyle\psi^{-}\left(v\right)$
$\displaystyle\geqslant\sup\left\\{s>0:C\cdot\pi\left(\varpi\geqslant\underline{\mathfrak{w}}\right)\cdot\tilde{\pi}_{*}\left(W\geqslant
s/\underline{\mathfrak{w}}\right)\geqslant v\right\\}$
$\displaystyle=\underline{\mathfrak{w}}\cdot\sup\left\\{s>0\colon\tilde{\pi}_{*}\left(W\geqslant
s\right)\geqslant C^{-1}\cdot
v/\pi\left(\varpi\geqslant\underline{\mathfrak{w}}\right)\right\\},$

that is, the inverse $\psi_{*}^{-}$ of $\tilde{\pi}_{*}$ provides the
functional form of a lower bound. Consider for example the scenario where
$Q_{*}$ is a ${\rm Pareto}\left(\alpha,x_{m}\right)$ with
$x_{m}=1-\alpha^{-1}$ and $\alpha>1$ to ensure that $\int w\cdot
Q_{*}\left({\rm d}w\right)=1$. In this specific case, the inverse
$\psi_{*}^{-1}$ is well-defined and it is shown in Appendix D.2 that for
$v>0$,

$\psi_{*}^{-1}\left(v\right)=x_{m}\cdot v^{-1/\left(\alpha-1\right)}\,.$

As a result, as the tails of the Pareto distribution become heavier (i.e.
$\alpha\downarrow 1$), the weak conductance profile
$\Phi_{\tilde{P}}^{\mathrm{W}}\left(v\right)$ vanishes more rapidly at $0$,
resulting in a slower convergence rate from Theorems 8 and 81 for a positive
chain.

######  Remark 83.

The results above require $q$ to be $\nu$-reversible, which covers the RWM
algorithm, but we suspect the results to hold more generally. A simple example
is the Independent MH algorithm. Assume that for some $\epsilon>0$, any
$x\in\mathsf{X}$, $\epsilon\leqslant\varpi\left(x\right)=\frac{{\rm
d}\pi}{{\rm d}q}\left(x\right)\geqslant\epsilon^{-1}$, let
$A_{s}:=\left\\{\left(x,w\right):w\geqslant
s\right\\}=\mathsf{X}\times\left\\{w\colon w\geqslant s\right\\}$ and define
the corresponding $\psi^{-}$ using (24). From Jensen’s inequality and these
assumptions, we have for $s\geqslant\epsilon^{-2}$ that

$\displaystyle\mathcal{E}\big{(}\tilde{P},\mathbf{1}_{A_{s}}\big{)}$
$\displaystyle=\int\pi$ $\displaystyle\left({\rm d}x\right)q\left(x,{\rm
d}y\right)\tilde{\pi}_{x}\left({\rm d}w\right)Q_{y}\left({\rm
d}u\right)\mathbf{1}\left\\{w\geqslant
s,u<s\right\\}\min\left\\{1,r\left(x,y\right)\frac{u}{w}\right\\}$
$\displaystyle\leqslant\int\pi$ $\displaystyle\left({\rm
d}x\right)\tilde{\pi}_{x}\left({\rm d}w\right)\mathbf{1}\left\\{w\geqslant
s\right\\}q\left(x,{\rm
d}y\right)\min\left\\{1,r\left(x,y\right)\cdot\frac{1}{w}\right\\}$
$\displaystyle\leqslant\int\pi$ $\displaystyle\left({\rm
d}x\right)\tilde{\pi}_{x}\left({\rm d}w\right)\mathbf{1}\left\\{w\geqslant
s\right\\}q\left(x,{\rm d}y\right)\epsilon^{-2}\cdot\frac{1}{w}$
$\displaystyle\leqslant\mu\left(A_{s}\right)/s\,,$

leading to identical conclusions as in Theorem 81 and Example 82.

######  Lemma 84.

Let $\tilde{P}$ be as in (23) and define, for any $s>0$,

$A_{s}:=\left\\{\left(x,w\right)\in\mathsf{X}\times\mathbb{R}_{+}\colon\varpi\left(x\right)\cdot
w\geqslant s\right\\}\,.$

Then for any $s>0$,

$\frac{\mathcal{E}\big{(}\tilde{P},\mathbf{1}_{A_{s}}\big{)}}{\mu\big{(}A_{s}\big{)}}\leqslant\frac{\int\pi\left(\mathrm{d}x\right)\tilde{\pi}_{x}\left(\varpi\left(x\right)W\geqslant
s\right)\int
q\left(x,\mathrm{d}y\right)\frac{\varpi(y)}{s}\tilde{\pi}_{y}\left(\varpi\left(y\right)W<s\right)}{\int\pi\left(\mathrm{d}x\right)\tilde{\pi}_{x}\left(\varpi\left(x\right)W\geqslant
s\right)}.$

###### Proof.

Let $s\geqslant 0$. We have that

$\displaystyle\mathcal{E}\big{(}\tilde{P},\mathbf{1}_{A_{s}}\big{)}$
$\displaystyle=\frac{1}{2}\int\nu\left(\mathrm{d}x\right)Q_{x}\left(\mathrm{d}w\right)q\left(x,\mathrm{d}y\right)Q_{y}\left(\mathrm{d}u\right)\mathbf{1}\left\\{s\leqslant\varpi\left(x\right)\cdot
w\right\\}$ $\displaystyle\hskip
113.81102pt\times\mathbf{1}\left\\{s>\varpi\left(y\right)u\right\\}\min\left\\{\varpi(x)w,\varpi(y)u\right\\}$
$\displaystyle=\frac{1}{2}\int\nu\left(\mathrm{d}x\right)Q_{x}\left(\mathrm{d}w\right)q\left(x,\mathrm{d}y\right)Q_{y}\left(\mathrm{d}u\right)\mathbf{1}\left\\{s\leqslant\varpi\left(x\right)w\right\\}$
$\displaystyle\hskip 28.45274pt\times\int_{0}^{\infty}{\rm
d}h\mathbf{1}\left\\{h\leq\varpi\left(x\right)w\right\\}\mathbf{1}\left\\{h\leq\varpi\left(y\right)u\right\\}$

For $x,y\in\mathsf{X}$,

$\int
Q_{x}\left(\mathrm{d}w\right)\mathbf{1}\left\\{h\leqslant\varpi\left(x\right)w\right\\}\mathbf{1}\left\\{s\leqslant\varpi\left(x\right)w\right\\}=\int_{0}^{\infty}Q_{x}\left(\varpi\left(x\right)W\geqslant
h\vee s\right),$

and

$\int
Q_{y}\left(\mathrm{d}u\right)\mathbf{1}\left\\{h\leqslant\varpi\left(y\right)u\right\\}\mathbf{1}\left\\{s>\varpi\left(y\right)u\right\\}=Q_{y}\left(h\leqslant\varpi\left(y\right)W<s\right).$

Further,

$\displaystyle\int_{0}^{\infty}Q_{x}$
$\displaystyle\left(\varpi\left(x\right)W\geqslant h\vee
s\right)Q_{y}\left(h\leqslant\varpi\left(y\right)W<s\right)\mathrm{d}h$
$\displaystyle=\mathbf{1}\left\\{h\leqslant
s\right\\}\int_{0}^{s}\,Q_{x}\left(\varpi\left(x\right)W\geqslant
s\right)Q_{y}\left(h\leqslant\varpi\left(y\right)W<s\right)\mathrm{d}h$
$\displaystyle=\mathbf{1}\left\\{h\leqslant
s\right\\}Q_{x}\left(\varpi\left(x\right)\cdot W\geqslant
s\right)\int_{0}^{s}Q_{y}\left(h\leqslant\varpi\left(y\right)W<s\right)\mathrm{d}h$

Changing the order of integration, for $h\leqslant s$, we obtain that

$\displaystyle\int_{0}^{s}Q_{y}\left(h\leqslant\varpi\left(y\right)W<s\right)\mathrm{d}h$
$\displaystyle=\int_{0}^{s}\int\mathbf{1}\left\\{h\leqslant\varpi\left(y\right)w<s\right\\}\mathrm{d}hQ_{y}\left({\rm
d}w\right)$
$\displaystyle=\int\varpi\left(y\right)w\mathbf{1}\left\\{\varpi\left(y\right)w<s\right\\}Q_{y}\left({\rm
d}w\right)$
$\displaystyle=\varpi\left(y\right)\tilde{\pi}_{y}\left(\varpi\left(y\right)W<s\right).$

Consequently, for $s>0$,

$\displaystyle\mathcal{E}\big{(}\tilde{P},\mathbf{1}_{A_{s}}\big{)}$
$\displaystyle=\int\nu$
$\displaystyle\left(\mathrm{d}x\right)q\left(x,\mathrm{d}y\right)\cdot
Q_{x}\left(\varpi\left(x\right)\cdot W\geqslant
s\right)\cdot\varpi\left(y\right)\cdot\tilde{\pi}_{y}\left(\varpi\left(y\right)\cdot
W<s\right)$ $\displaystyle=\int\pi$
$\displaystyle\left(\mathrm{d}x\right)q\left(x,\mathrm{d}y\right)\cdot\frac{1}{\varpi\left(x\right)}\cdot
Q_{x}\left(\varpi\left(x\right)\cdot W\geqslant
s\right)\cdot\varpi\left(y\right)\cdot\tilde{\pi}_{y}\left(\varpi\left(y\right)\cdot
W<s\right)$ $\displaystyle\leqslant\int\pi$
$\displaystyle\left(\mathrm{d}x\right)q\left(x,\mathrm{d}y\right)\cdot\frac{1}{s}\cdot\tilde{\pi}_{y}\left(\varpi\left(x\right)\cdot
W\geqslant
s\right)\cdot\varpi\left(y\right)\cdot\tilde{\pi}_{y}\left(\varpi\left(y\right)\cdot
W<s\right),$

because

$\displaystyle\frac{1}{\varpi\left(x\right)}Q_{x}\left(\varpi\left(x\right)\cdot
W\geqslant s\right)$
$\displaystyle=\int\frac{1}{\varpi\left(x\right)}\mathbf{1}\left\\{\varpi\left(x\right)\cdot
w\geqslant s\right\\}\cdot Q_{x}\left({\rm d}w\right)$
$\displaystyle=\int\frac{1}{\varpi\left(x\right)\cdot
w}\mathbf{1}\left\\{\varpi\left(x\right)\cdot w\geqslant s\right\\}\cdot
w\cdot Q_{x}\left({\rm d}w\right)\,.$ $\displaystyle\leqslant
s^{-1}\cdot\int\mathbf{1}\left\\{\varpi\left(x\right)\cdot w\geqslant
s\right\\}\cdot w\cdot Q_{x}\left({\rm d}w\right)\,.$

The result follows. ∎

##  5 Applications to RWM on heavy-tailed targets

We assume now that $\pi$ has density $\varpi=\frac{{\rm d}\pi}{{\rm
d}\lambda}$ with respect to some $\sigma$-finite measure $\lambda$, and that
$Q$ is a $\lambda$-reversible Markov kernel. A Metropolis Markov kernel $P$
may be defined [28] as

$P\left(x,A\right)=\int_{A}Q\left(x,{\rm
d}y\right)\cdot\alpha\left(x,y\right)+{\bf
1}_{A}\left(x\right)\cdot\bar{\alpha}\left(x\right),\qquad
x\in\mathsf{E},A\in\mathscr{E}.$ (25)

where for $x,y\in\mathsf{E}$,

$\alpha\left(x,y\right):=1\wedge\frac{\varpi\left(y\right)}{\varpi\left(x\right)},\quad\bar{\alpha}\left(x\right):=1-\alpha\left(x\right),\quad\alpha\left(x\right):=\int_{\mathsf{E}}Q\left(x,{\rm
d}y\right)\cdot\alpha\left(x,y\right).$ (26)

######  Example 85.

In many applications, $Q\left(x,\cdot\right)$ is a multivariate normal
distribution with mean $x$ and covariance matrix $\sigma^{2}\cdot I_{d}$: for
some $\sigma>0$,

$Q\left(x,A\right)=\int{\bf 1}_{A}\left(x+\sigma\cdot z\right)\,{\cal
N}\left({\rm d}z;0,I_{d}\right),\qquad x\in\mathsf{E},A\in\mathscr{E}.$

Since $Q$ is reversible w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$, one
can take $\lambda\left({\rm d}x\right)={\rm d}x$ and consider target
distributions $\pi$ with densities w.r.t. the Lebesgue measure. This defines
the Random-Walk Metropolis (RWM) Markov kernel. We remark that various other
proposals can also readily be analysed; see [2, Appendix C] for relevant
details.

The following result can be used to establish the close coupling condition of
Definition 73 for Metropolis kernels, which we will use in the sequel to
analyze the RWM chain.

######  Lemma 86.

(Lemma 19 of [2]) Let $Q$ be a $\lambda$-reversible Markov kernel, where
$\lambda\gg\pi$ is a $\sigma$-finite measure, $P$ be the $\pi$-reversible
Metropolis kernel with proposal $Q$ defined by (25) and
$\alpha_{0}:=\inf_{x\in\mathsf{E}}\alpha\left(x\right)$. Then

$\left\|P\left(x,\cdot\right)-P\left(y,\cdot\right)\right\|_{{\rm
TV}}\leqslant\left\|Q\left(x,\cdot\right)-Q\left(y,\cdot\right)\right\|_{{\rm
TV}}+1-\alpha_{0},\qquad x,y\in\mathsf{E}.$

We now assume Assumption 68 to hold, that is
$\varpi\propto\exp\left(-U\right)$ with $\lambda$ the Lebesgue measure and $Q$
is as in Example 85. It is standard to deduce by [8, Lemma 3.1] that $P$ is a
positive Markov kernel for this particular $Q$. We note the following useful
expression

$\alpha\left(x\right)=\int{\cal N}\left({\rm
d}z;0,I_{d}\right)\cdot\min\left\\{1,\exp\left(-\left(U\left(x+\sigma\cdot
z\right)-U\left(x\right)\right)\right)\right\\}\,.$ (27)

###  5.1 Mixing times of RWM for $L$-smooth targets

For the purposes of obtaining explicit bounds and matching negative results
with dimension, we impose the following further assumption about $\pi$, noting
that Assumption 68 is already in force. As established in Lemma 39 of [2],
this condition can be weakened to accommodate more general quantitative
continuity assumptions; we do not linger on this point here.

######  Assumption 87.

For some $L\in\left(0,\infty\right)$, $U$ is $L$-smooth:

$U\left(x+h\right)-U\left(x\right)-\left\langle\nabla
U\left(x\right),h\right\rangle\leqslant\frac{L}{2}\cdot\left|h\right|^{2},\qquad
x,h\in\mathsf{E}.$

######  Lemma 88.

([2, Lemma 37, Corollary 40]) Suppose that the potential
$U:\mathbb{R}^{d}\to\mathbb{R}$ is $L$-smooth, and let $P$ be the RWM kernel
with target distribution $\pi=\exp\left(-U\right)$ and step-size
$\sigma=\varsigma\cdot\left(L\cdot d\right)^{-1/2}$. It then holds that $P$ is
$\left(\left|\cdot\right|,\alpha_{0}\cdot\sigma,\frac{1}{2}\cdot\alpha_{0}\right)$-close
coupling with
$\alpha_{0}\geqslant\frac{1}{2}\cdot\exp\left(-\frac{1}{2}\cdot\varsigma^{2}\right)$.

######  Theorem 89.

Suppose that $\pi$ is a probability measure on $\mathbb{R}^{d}$ with regular,
convex isoperimetric minorant $\tilde{I}_{\pi}$, and that the potential
$U=-\log\pi$ is $L$-smooth. Let $P$ be the RWM kernel with target distribution
$\pi$ and step-size $\sigma=\varsigma\cdot\left(L\cdot d\right)^{-1/2}$.
Assume also that
$2\cdot\varsigma\cdot\left(L^{-1/2}\cdot\tilde{I}_{\pi}\left(\frac{1}{4}\right)\right)\leqslant
d^{1/2}$. It then holds that

  1. a).

$P$ has a positive WCP, satisfying for $v\in\left(0,\frac{1}{2}\right]$,

$\Phi_{P}^{\mathrm{W}}\left(v\right)\geqslant
2^{-6}\cdot\varsigma\cdot\exp\left(-\varsigma^{2}\right)\cdot\left(L\cdot
d\right)^{-1/2}\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot v}.$

  2. b).

$P$ satisfies a $K^{*}$_–_ WPI, with $K^{*}$ lower-bounded as

$K^{*}\left(v\right)\geqslant
2^{-11}\cdot\varsigma^{2}\cdot\exp\left(-2\cdot\varsigma^{2}\right)\cdot\left(L\cdot
d\right)^{-1}\cdot\frac{\tilde{I}_{\pi}\left(2^{-3}\cdot
v\right)^{2}}{2^{-3}\cdot v}.$

  3. c).

For bounded functions $f$, writing
$v_{n}:=\left\|P^{n}f-\mu\left(f\right)\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\in\left[0,\frac{1}{4}\right]$,
it holds that

$n\leqslant
2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot L\cdot
d\cdot\int_{2^{-3}\cdot v_{n}}^{2^{-3}\cdot
v_{0}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}.$

  4. d).

Writing $\nu$ for the initial law of the chain,
$u:=\left\|\frac{\mathrm{d}\nu}{\mathrm{d}\pi}\right\|_{\mathrm{osc}}^{2}<\infty$,
then in order to guarantee that

$\chi^{2}\left(\nu P^{n},\pi\right)\leqslant\varepsilon_{{\rm
Mix}}\quad\text{for some }\varepsilon_{{\rm Mix}}>0,$

it suffices that

$n\geqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot
L\cdot d\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}\right).$

###### Proof.

For a), we aim to apply Corollary 75, whose notation we use. We have by Lemma
88 that

$\frac{1}{2}\cdot\delta\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot
v}=\frac{1}{2}\alpha_{0}\cdot\sigma\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot v}\leqslant\frac{1}{2}\cdot\varsigma\left(L\cdot
d\right)^{-1/2}\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{4}\right)}{\frac{1}{4}}\leqslant
1,$

and then Lemma 88 yields the lower bound

$\displaystyle\frac{1}{2}\alpha_{0}\cdot\sigma\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot v}$
$\displaystyle\geqslant\frac{1}{2^{2}}\cdot\exp\left(-\frac{1}{2}\cdot\varsigma^{2}\right)\cdot\varsigma\left(L\cdot
d\right)^{-1/2}\cdot\frac{\tilde{I}_{\pi}\left(\frac{1}{2}\cdot
v\right)}{\frac{1}{2}\cdot v}.$

Again from Lemma 88 we have
$\varepsilon=\frac{1}{2}\cdot\alpha_{0}\geqslant\frac{1}{4}\cdot\exp\left(-\frac{1}{2}\cdot\varsigma^{2}\right)$
and we conclude with Corollary 75 . For b) we use Theorem 35. Statements c)
and d) follow from Theorem 8 and Theorem 76. ∎

######  Remark 90.

The condition
$2\cdot\varsigma\cdot\left(L^{-1/2}\cdot\tilde{I}_{\pi}\left(\frac{1}{4}\right)\right)\leqslant
d^{1/2}$ is extremely weak; in the ‘well-conditioned’ log-concave case, the
quantity $L^{-1/2}\cdot\tilde{I}_{\pi}\left(\frac{1}{4}\right)$ is (up to
constant factors) equal to the reciprocal of the condition number (this
follows from e.g. Lemma 27 of [2]), which is always less than $1$. Noting that
our bounds are almost all optimised by taking
$\varsigma\in\Theta\left(1\right)$, we expect this condition to almost always
be satisfied in practice, particularly for high-dimensional problems.

######  Remark 91.

We comment briefly on the assumptions made in preparation for this theorem.
The assumption of $L$-smoothness is conventional in this area, and functions
largely as a simple way of ensuring that the acceptance rates of the RWM chain
are well-controlled. Section 4.1 of [2] discusses some relaxations of this
assumption which accommodate potentials with lower regularity. Control of the
isoperimetric profile is crucial to our approach, and appears to give quite
honest bounds (“given an accurate bound on $I_{\pi}$, one expects to reproduce
the correct rate of convergence”). It is not a strong assumption on a
probability measure to assume that it has a regular isoperimetric profile (or
minorant), but in specific practical scenarios, characterising the profile
accurately can be challenging. For applying our results in practice, the main
barrier is thus to obtain good control of the isoperimetric profile.

###  5.2 Mixing times: examples

In this paper we consider heavy-tailed distributions for which the constant
$L$ may be dimension dependent, which is to be contrasted with the light-
tailed distributions considered in [2]. In the following, $d\in\mathbb{N}$
denotes the dimension of the ambient space $\mathsf{E}=\mathbb{R}^{d}$.

######  Example 92.

For the standard multivariate Student’s t-distribution in dimension $d$ with
$\tau>0$ degrees of freedom, one has

$U(x)=\frac{d+\tau}{2}\cdot\log\big{(}\tau+\left|x\right|^{2}\big{)}\,,$

and can take $L=1+d\cdot\tau^{-1}$; see Appendix D.1 for details. Naturally
with many degrees of freedom (i.e. $\tau\gtrsim d$), $L$ becomes less
dimension dependent, as the Student’s t-distribution tends towards a Gaussian
limit.

######  Example 93.

For the $d$-variate distribution with potential given by

$U\left(x\right)=\left(\tau+\left|x\right|^{2}\right)^{\alpha/2}\text{ for
}\alpha\in\left(0,1\right),$

one can take $L=\alpha\cdot\tau^{-\left(1-\frac{\alpha}{2}\right)}$; see
Appendix D.1 for details.

For some initial law $\nu\ll\pi$ we let
$u=\left\|\mathrm{d}\nu/\mathrm{d}\pi\right\|_{\mathrm{osc}}^{2}$ and let
$\varsigma$ denote a non-dimensionalised step-size which should be treated as
being of constant order. We will use ‘warm start’ to denote an initialisation
for which $u\in\mathcal{O}\left(1\right)$, and ‘feasible start’ to denote an
initialisation for which $\log u\in\mathcal{O}\left(d\right)$; one can see
that this is a reasonable scaling by considering product-form $\nu$ and $\pi$,
i.e. if $\nu$, $\pi$ are mutually absolutely continuous and satisfy
$c_{-}\leqslant\frac{\mathrm{d}\nu}{\mathrm{d}\pi}\leqslant c_{+}$ for some
$0<c_{-}<1<c_{+}<\infty$, then direct computations yield that

$\left\|\frac{\mathrm{d}\nu^{\otimes d}}{\mathrm{d}\pi^{\otimes
d}}\right\|_{\mathrm{osc}}^{2}=c_{+}^{d}-c_{-}^{d}\geqslant
c_{+}^{d}-1\in\exp\left(\Theta\left(d\right)\right)\,.$

Recalling Theorem 8, one sees that for a Markov chain initialised at
$X_{0}\sim\nu$, it suffices to take $n\geqslant
n\left(\varepsilon_{\mathrm{Mix}}\cdot
u^{-1};\left\|\cdot\right\|_{\mathrm{osc}}^{2}\right)$ in order to ensure that
$\chi^{2}\left(\nu P^{n},\pi\right)\leqslant\varepsilon_{\mathrm{Mix}}$. As
such, subsequent mixing time estimates will be presented in terms of bounds on
this quantity, which we shorten to
$n_{\star}\left(\varepsilon_{\mathrm{Mix}};u\right)$ for convenience.

######  Example 94 (Example 92 ctd.).

For $\tau>0$, let
$U\left(x\right)=\frac{d+\tau}{2}\cdot\log\big{(}\tau+\left|x\right|^{2}\big{)}$.
One can then take $L=1+d\cdot\tau^{-1}$,
$\tilde{I}_{\pi}\left(p\right)=c\left(d,\tau\right)\cdot d^{-1/2}\cdot
p^{1+1/\tau}$, where $c\left(d,\tau\right)$ is bounded uniformly in $d$ for
suitably ‘high-dimensional’ problems; see the statement of Lemma 95 for a
precise statement. With $\sigma=\varsigma\cdot\left(L\cdot d\right)^{-1/2}$,
one then obtains

$\displaystyle n_{\star}\left(\varepsilon_{\mathrm{Mix}};u\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot
L\cdot d\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d^{2}}{c\left(d,\tau\right)^{2}}\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm
Mix}}\cdot u^{-1}}^{2^{-5}}\frac{\mathrm{d}v}{v^{1+2/\tau}}\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d^{2}}{c\left(d,\tau\right)^{2}}\cdot\frac{\tau}{2}\cdot\left(2^{-3}\cdot\varepsilon_{{\rm
Mix}}\cdot u^{-1}\right)^{-2/\tau}$
$\displaystyle=1+2^{13+6/\tau}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d^{2}}{c\left(d,\tau\right)^{2}}\cdot\tau\cdot\left(\frac{u}{\varepsilon_{{\rm
Mix}}}\right)^{2/\tau}$
$\displaystyle=\mathcal{O}\left(d^{2}\cdot\left(\frac{u}{\varepsilon_{{\rm
Mix}}}\right)^{2/\tau}\right).$

Thus, under a warm start, the mixing time scales like
$d^{2}\cdot\varepsilon_{{\rm Mix}}^{-2/\tau}$, whereas from a feasible start,
it takes $\exp\left(\mathcal{O}\left(d\right)\right)$ to reach
$\varepsilon_{{\rm Mix}}\asymp 1$.

######  Lemma 95.

Let $\pi$ be as defined in Example 92. Assume that the parameters
$\left(d,\tau\right)\in\mathbb{N}\times\left(0,\infty\right)$ satisfy
$\sqrt{\frac{2}{\min\left\\{d,\tau\right\\}}}\leqslant\xi$ for some
$\xi\in\left(0,1\right)$. One can then lower bound the isoperimetric profile
of $\pi$ as

$\tilde{I}_{\pi}\left(p\right)\geqslant
c\left(\tau\right)\cdot\left(\frac{1-\xi}{1+\xi}\right)^{1/2}\cdot
d^{-1/2}\cdot p^{1+1/\tau},$

where the constant $c\left(\tau\right)>0$ is independent of the dimension.

###### Proof.

By [9, Theorem 1.2], it is known that one can minorise

$\tilde{I}_{\pi}\left(p\right)\geqslant\frac{c\left(\tau\right)}{\mathrm{med}\left(\left|X\right|\right)}\cdot
p^{1+\frac{1}{\tau}}\,,$

for some universal constant $c\left(\tau\right)>0$. It thus remains to control
$\mathrm{med}\left(\left|X\right|\right)$. For $X\sim\pi$, one can verify that
with $S:=\tau^{-1}\cdot\left|X\right|^{2}$ and $B:=S/\left(1+S\right)$ then it
holds that $B\overset{\mathrm{d}}{=}\mathrm{Beta}\left(d/2,\tau/2\right)$. By
a standard result for random variables with finite second order moment, we
have for $Z\sim\mathrm{Beta}(\alpha,\beta)$ with $\alpha,\beta>0$,
$\big{|}\mathbb{E}\left[Z\right]-\mathrm{med}\left(Z\right)\big{|}^{2}\leqslant\mathbb{E}\left[Z^{2}\right]-\mathbb{E}\left[Z\right]^{2}$,
so that

$\displaystyle\mathrm{med}\left(Z\right)$
$\displaystyle\leqslant\frac{\alpha}{\alpha+\beta}+\sqrt{\frac{\alpha\cdot\beta}{(\alpha+\beta)^{3}}}.$

Translating this to our setting and noting that the median is covariant under
monotonic maps, one then computes that

$\displaystyle\mathrm{med}\left(\left|X\right|\right)$
$\displaystyle\leqslant\left(\tau\cdot\frac{d+\sqrt{\frac{2\cdot
d\cdot\tau}{\left(d+\tau\right)}}}{\tau-\sqrt{\frac{2\cdot
d\cdot\tau}{\left(d+\tau\right)}}}\right)^{1/2}.$

Compute that (using that
$d\cdot\tau=\min\left\\{d,\tau\right\\}\cdot\max\left\\{d,\tau\right\\}$)

$\displaystyle\sqrt{\frac{2\cdot d\cdot\tau}{\left(d+\tau\right)}}$
$\displaystyle=\min\left\\{d,\tau\right\\}\cdot\sqrt{\frac{\max\left\\{d,\tau\right\\}}{\left(d+\tau\right)}\cdot\frac{2}{\min\left\\{d,\tau\right\\}}}\leqslant\min\left\\{d,\tau\right\\}\cdot\xi,$

where the final inequality follows from the assumption that
$\xi\geqslant\sqrt{\frac{2}{\min\left\\{d,\tau\right\\}}}$. We hence deduce
that

$\frac{d+\sqrt{\frac{2\cdot
d\cdot\tau}{\left(d+\tau\right)}}}{\tau-\sqrt{\frac{2\cdot
d\cdot\tau}{\left(d+\tau\right)}}}\leqslant\frac{d+d\cdot\xi}{\tau-\tau\cdot\xi}=\frac{1+\xi}{1-\xi}\cdot\frac{d}{\tau},$

from which the result follows. ∎

######  Example 96 (Product form of Example 92).

For $\eta>0$, let $U\left(x\right)=\sum_{i\in\llbracket
d\rrbracket}\frac{1+\eta}{2}\cdot\log\left(1+\left|x_{i}\right|^{2}\right)$.
One can then take $L=1+\eta$, $I_{\pi}\left(p\right)\geqslant
c\left(\eta\right)\cdot d^{-1/\eta}\cdot p^{1+1/\eta}$ . With
$\sigma=\varsigma\cdot\left(L\cdot d\right)^{-1/2}$, one then obtains

$\displaystyle n_{\star}\left(\varepsilon_{\mathrm{Mix}};u\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot
L\cdot d\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d}{c\left(\eta\right)^{2}}\cdot
d^{2/\eta}\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{\mathrm{d}v}{v^{1+2/\eta}}\right)$
$\displaystyle=1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d}{c\left(\eta\right)^{2}}\cdot
d^{2/\eta}\cdot\frac{\eta}{2}\cdot\left(2^{-3}\cdot\varepsilon_{{\rm
Mix}}\cdot u^{-1}\right)^{-2/\eta}$
$\displaystyle=1+2^{13+6/\eta}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot\frac{L\cdot
d^{1+2/\eta}}{c\left(\eta\right)^{2}}\cdot\eta\cdot\left(\frac{u}{\varepsilon_{{\rm
Mix}}}\right)^{2/\eta}$
$\displaystyle=\mathcal{O}\left(d^{1+2/\eta}\cdot\left(\frac{u}{\varepsilon_{{\rm
Mix}}}\right)^{2/\eta}\right).$

Thus, under a warm start, the mixing time scales like
$d^{1+2/\eta}\cdot\varepsilon_{{\rm Mix}}^{-2/\eta}$, whereas from a feasible
start, it takes $\exp\mathcal{O}\left(d\right)$ to reach $\varepsilon_{{\rm
Mix}}\asymp 1$.

######  Example 97 (Product form of Example 93).

For $\eta\in\left(0,1\right)$, $\tau>0$, let
$U\left(x\right)=\sum_{i\in\llbracket
d\rrbracket}\left(\tau+\left|x_{i}\right|^{2}\right)^{\eta/2}$. One can take
$L=\eta\cdot\tau^{-\left(1-\frac{\eta}{2}\right)}$,
$I_{\pi}\left(p\right)\geqslant c\left(\eta,\tau\right)\cdot
p\cdot\left(\log\left(\frac{d}{p}\right)\right)^{-\left(1/\eta-1\right)}$.
With $\sigma=\varsigma\cdot\left(L\cdot d\right)^{-1/2}$, one then obtains

$\displaystyle n_{\star}\left(\varepsilon_{\mathrm{Mix}};u\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\varsigma^{-2}\cdot\exp\left(2\cdot\varsigma^{2}\right)\cdot
L\cdot d\cdot\left(\int_{2^{-3}\cdot\varepsilon_{{\rm Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{v\,\mathrm{d}v}{\tilde{I}_{\pi}\left(v\right)^{2}}\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\frac{\exp\left(2\cdot\varsigma^{2}\right)}{\varsigma^{2}}\cdot\frac{L\cdot
d}{c\left(\eta,\tau\right)^{2}}\left(\int_{2^{-3}\cdot\varepsilon_{{\rm
Mix}}\cdot
u^{-1}}^{2^{-5}}\frac{\left(\log\left(\frac{d}{v}\right)\right)^{2\cdot\left(1/\eta-1\right)}}{v}\,\mathrm{d}v\right)$
$\displaystyle=1+2^{14}\cdot\frac{\exp\left(2\cdot\varsigma^{2}\right)}{\varsigma^{2}}\cdot\frac{L\cdot
d}{c\left(\eta,\tau\right)^{2}}\left(\int_{\log\left(2^{5}\cdot
d\right)}^{\log\left(2^{3}\cdot\frac{d\cdot u}{\varepsilon_{{\rm
Mix}}}\right)}t^{2\cdot\left(1/\eta-1\right)}\,\mathrm{d}t\right)$
$\displaystyle\leqslant
1+2^{14}\cdot\frac{\exp\left(2\cdot\varsigma^{2}\right)}{\varsigma^{2}}\cdot\frac{L\cdot
d}{c\left(\eta,\tau\right)^{2}}\cdot\frac{\eta}{2-\eta}\left(\log\left(2^{3}\cdot\frac{d\cdot
u}{\varepsilon_{{\rm Mix}}}\right)\right)^{2/\eta-1}$
$\displaystyle=\mathcal{O}\left(d\cdot\left(\log\left(\frac{d\cdot
u}{\varepsilon_{{\rm Mix}}}\right)\right)^{2/\eta-1}\right).$

Thus, under a warm start, the mixing time scales like
$d\cdot\left(\log\left(\frac{d}{\varepsilon_{{\rm
Mix}}}\right)\right)^{2/\eta-1}$, whereas from a feasible start, it takes
$d^{2/\eta}$ to reach $\varepsilon_{{\rm Mix}}\asymp 1$.

Relative to the light-tailed setting, we see that in the heavy-tailed setting,
we pay a more dramatic price for poor initialisation. In particular, in
Example 96, while the complexity of mixing to within a given error
$\varepsilon_{\mathrm{Mix}}$ scales only polynomially with both $d$ and
$\varepsilon_{\mathrm{Mix}}$ under a warm start (i.e.
$u_{0}\in\mathcal{O}\left(1\right)$), the dependence on the initial $u_{0}$ is
a polynomial in $d$ of potentially high degree in the case of quasi-
exponential tails, and exponential in $d$ in the case of polynomial tails.
This highlights explicitly that initialisation is of particular concern in the
heavy-tailed setting. We are not aware of explicit practical recommendations
on how to ameliorate this issue in practice.

###  5.3 Asymptotic variance: examples

Following the calculations in [3], we note that for sufficiently nice (i.e.
$\Psi$-finite) functions $f$ and sufficiently fast-mixing kernels $P$, we can
estimate the asymptotic variance of estimates of $\pi\left(f\right)$ which are
obtained as ergodic averages along the Markov chain. In particular, following
Theorem 17 of [3], for $\mu$-reversible kernels $P$ such that $P^{*}P$
satisfies a $\left(\Psi,K^{*}\right)$_–_ WPI, if $\mathrm{id}/K^{*}$ is
integrable around $0$, then there holds the general bound

$\mathrm{var}\left(P,f\right)\leqslant 4\cdot\Psi\left(f\right)\cdot
B\left(\frac{\left\|f\right\|_{2}^{2}}{\Psi\left(f\right)}\right)\text{ with
}B\left(v\right)=\int_{0}^{v}\frac{w}{K^{*}\left(w\right)}\,\mathrm{d}w\,.$

######  Remark 98.

In the context of RWM, derivations in Theorem 89 tell us that

$K^{*}\left(w\right)\gtrsim\left(L\cdot
d\right)^{-1}\cdot\frac{\tilde{I}_{\pi}\left(w\right)^{2}}{w}\implies\frac{w}{K^{*}\left(w\right)}\lesssim
L\cdot d\cdot\left(\frac{w}{\tilde{I}_{\pi}\left(w\right)}\right)^{2}.$

We thus intuit that if $\tilde{I}_{\pi}\left(w\right)\ll w^{3/2}$ at $0$, then
the integral which defines $B$ will not be proper, thus precluding a control
of the asymptotic variance with the present techniques.

######  Example 99 (Example 92 ctd.).

Consider the $d$-variate Student-t distribution with $\tau$ degrees of
freedom. For simplicity, assume that $\tau\geqslant d\geqslant 2$. Given the
estimate $I_{\pi}\left(p\right)\gtrsim d^{-1/2}\cdot p^{1+1/\tau}$, it follows
that $\frac{w}{K^{*}\left(w\right)}\lesssim d^{2}\cdot w^{-2/\tau}$ and hence
that $B\left(v\right)\lesssim d^{2}\cdot v^{1-2/\tau}$; one then concludes
that $\mathrm{var}\left(P,f\right)\lesssim
d^{2}\cdot\Psi\left(f\right)^{2/\tau}\cdot\left\|f\right\|_{2}^{2\cdot\left(1-2/\tau\right)}$.

######  Example 100 (Product form of Example 92).

Consider now the product-form target with potential
$U\left(x\right)=\sum_{i\in\llbracket
d\rrbracket}\frac{1+\eta}{2}\cdot\log\left(1+\left|x_{i}\right|^{2}\right)$
for $\eta>0$. Given the estimate $I_{\pi}\left(p\right)\gtrsim
d^{-1/\eta}\cdot p^{1+1/\eta}$, it follows that
$\frac{w}{K^{*}\left(w\right)}\lesssim d^{1+2/\eta}\cdot w^{-2/\eta}$ and
hence that $B\left(v\right)\lesssim d^{1+2/\eta}\cdot v^{1-2/\eta}$; one then
concludes that $\mathrm{var}\left(P,f\right)\lesssim
d^{1+2/\eta}\cdot\Psi\left(f\right)^{2/\eta}\cdot\left\|f\right\|_{2}^{2\cdot\left(1-2/\eta\right)}$.

######  Example 101 (Product form of Example 93).

Finally, consider the product-form target distribution with potential
$U\left(x\right)=\sum_{i\in\llbracket
d\rrbracket}\left(\tau+\left|x_{i}\right|^{2}\right)^{\eta/2}$ for
$\eta\in\left(0,1\right)$, $\tau>0$. Given the estimate
$I_{\pi}\left(p\right)\gtrsim
p\cdot\left(\log\left(\frac{d}{p}\right)\right)^{-\left(1/\eta-1\right)}$, it
follows that

$\frac{w}{K^{*}\left(w\right)}\lesssim
d\cdot\left(\log\left(\frac{d}{w}\right)\right)^{2\cdot\left(1/\eta-1\right)}\implies
B\left(v\right)\lesssim d\cdot
v\cdot\left(\log\left(\frac{d}{v}\right)\right)^{2\cdot\left(1/\eta-1\right)}$

and one then concludes that

$\mathrm{var}\left(P,f\right)\lesssim
d\cdot\left\|f\right\|_{2}^{2}\cdot\left(\log\left(\frac{d\cdot\Psi\left(f\right)}{\left\|f\right\|_{2}^{2}}\right)\right)^{2\cdot\left(1/\eta-1\right)}.$

##  6 Discussion

We now discuss relations to existing literature. WPIs were formally introduced
in [38], building on ideas of [26], to study subgeometric convergence rates of
diffusion processes, and have been studied extensively in the continuous-time
setting in works such as [14, 7]. More recently, there has been a renewed
interest in WPIs for the study of computational statistical algorithms, often
in the context of heavy-tailed distributions; see [1, 31].

In terms of prior results on the convergence to equilibrium of RWM on heavy-
tailed target measures, prior work has been largely qualitative; see e.g. [21,
23, 22, 18] based on the drift and minorisation approach. Optimal scaling-type
results (as in [32, 33] where scaling is with respect to $d$ for specific
classes of targets) also offer some insights on this setting [35], though
require some care in interpretation. In particular, in the heavy-tailed case,
it appears that convergence of low-dimensional marginal distributions may
happen appreciably faster than joint convergence, as suggested by our results.

As in our previous work [2], the quality of our convergence estimates for RWM
is contingent on the availability of accurate estimates on the isoperimetric
profile of the target measure. While in the log-concave case, many techniques
are currently available for this task, the situation remains less developed in
the heavy-tailed setting. In this work, we have relied heavily on the
impressively general results of [9, 14], but one suspects that in many
specific instances, it should be possible to improve these estimates
substantially, particularly in terms of dimension-dependence. We would thus
welcome additional developments on the study of heavy-tailed isoperimetry, and
heavy-tailed functional inequalities more broadly.

##  Appendix A Notation

  * •

We will write $\mathbb{N}=\left\\{1,2,\dots\right\\}$ for the set of natural
numbers, $\mathbb{N}_{0}:=\mathbb{N}\cup\left\\{0\right\\}$, and
$\mathbb{R}_{+}=\left(0,\infty\right)$ for positive real numbers.

  * •

For $d\in\mathbb{N}$, write $\llbracket
d\rrbracket=\left\\{1,2,\cdots,d\right\\}$.

  * •

For $f\colon\mathbb{R}\rightarrow\mathbb{R}$ we say that $f$ is increasing
(resp. decreasing) if $a<b\implies f\left(a\right)\leqslant f\left(b\right)$
(resp. $a<b\implies f\left(a\right)\geqslant f\left(b\right)$) and is strictly
increasing (resp. strictly decreasing) if $a<b\implies
f\left(a\right)<f\left(b\right)$ (resp. $a<b\implies
f\left(a\right)>f\left(b\right)$).

  * •

For vectors $v\in\mathbb{R}^{d}$, write $\left|v\right|$ for the Euclidean
norm of $v$. For positive semi-definite matrices $M\in\mathbb{R}^{d\times d}$,
write
$\left\|M\right\|_{\mathrm{spec}}=\sup\left\\{v^{\top}Mv:\left|v\right|\leqslant
1\right\\}$ for the spectral norm of $M$.

  * •

We adopt the following $\mathcal{O}$ (resp. $\Omega$) notation to indicate
when functions grow no faster than (resp. no slower than) other functions. For
$a\in\mathbb{R}\cup\left\\{\infty\right\\}$,

    * –

If $f(x)\in\mathcal{O}\left(g\left(x\right)\right)$ as $x\to a$, this means
that $\underset{x\to a}{\lim\sup}\left|\frac{f(x)}{g(x)}\right|<\infty$. When
$a=+\infty$, then we may drop explicit mention of $a$.

    * –

If $f(x)\in\Omega\left(g\left(x\right)\right)$ as $x\to a$, this means that
$\underset{x\to a}{\lim\inf}\left|\frac{f(x)}{g(x)}\right|>0$. In particular
$f\in\mathcal{O}\left(g\right)\iff g\in\Omega\left(f\right)$.

Outside of specific examples, we will be working throughout on a general
measurable space $\left(\mathsf{E},\mathscr{E}\right)$.

  * •

For $\mu$ a probability measure on $\left(\mathsf{E},\mathscr{E}\right)$ and
$Z\sim\mu$, we write $\mathbb{E}\left[Z\right]$ for the expectation of $Z$.

  * •

For $\left\\{Z_{n},n\in\mathbb{N}\right\\}$ a sequence of random variables, we
let $Z_{n}\overset{L}{\rightarrow}Z$ indicate convergence in distribution to
the random variable $Z$.

  * •

For a set $A\in\mathscr{E}$, its complement in $\mathsf{E}$ is denoted by
$A^{\complement}$. We denote the corresponding indicator function by
$\mathbf{1}_{A}:\mathsf{E}\to\left\\{0,1\right\\}$.

  * •

We assume that $\left(\mathsf{E},\mathscr{E}\right)$ is equipped with a
probability measure $\mu$, and write $\mathrm{L}^{2}\left(\mu\right)$ for the
Hilbert space of (equivalence classes of) real-valued $\mu$–square-integrable
measurable functions with inner product

$\langle
f,g\rangle=\int_{\mathsf{E}}f\left(x\right)g\left(x\right)\,\mathrm{d}\mu\left(x\right)\,,$

and corresponding norm $\left\|\cdot\right\|_{2,\mu}$, and if there is no
ambiguity, we may just write $\left\|\cdot\right\|_{2}$. We write
$\mathrm{L}^{2}_{0}\left(\mu\right)$ for the set of functions
$f\in\mathrm{L}^{2}\left(\mu\right)$ which also satisfy $\mu\left(f\right)=0$.

  * •

More generally, for $p\in\left[1,\infty\right)$, we write
$\mathrm{L}^{p}\left(\mu\right)$ for the Banach space of real-valued
measurable functions with finite $p$-norm,
$\left\|f\right\|_{p}:=\left(\int_{\mathsf{E}}\left|f\right|^{p}\,\mathrm{d}\mu\right)^{1/p}$,
and $\mathrm{L}_{0}^{p}\left(\mu\right)$ for
$f\in\mathrm{L}^{p}\left(\mu\right)$ with $\mu\left(f\right)=0$.

  * •

For $g\in\mathrm{L}^{2}\left(\mu\right)$, let
$\mathrm{var}_{\mu}\left(g\right):=\left\|g-\pi\left(g\right)\right\|_{2}^{2}$.
We write $\mathrm{L}^{2}_{0}\left(\mu\right)$ for the set of functions
$f\in\mathrm{L}^{2}\left(\mu\right)$ which also satisfy $\mu\left(f\right)=0$.

  * •

For $\nu\in\mathcal{P}\left(\mathsf{E}\right)$ such that $\nu\ll\mu$ and
$f=\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\in\mathrm{L}^{2}\left(\mu\right)$,
write $\chi^{2}\left(\nu,\mu\right)$ for $\mathrm{var}_{\mu}\left(f\right)$.

  * •

For $g\in\mathrm{L}^{1}\left(\mu\right)$, let
$\mathrm{var}_{\mu}^{\left(1\right)}\left(g\right):=\inf_{c\in\mathbb{R}}\left\|g-c\right\|_{1}$,
sometimes referred to as mean absolute deviation, and we denote by
$\mathrm{med}_{\mu}\left(g\right)$ any $c\in\mathbb{R}$ which attains this
infimum; we will sometimes drop the subscript $\mu$ when it is clear from
context.

  * •

For a measurable function $f:\mathsf{E}\to\mathbb{R}$, let
$\left\|f\right\|_{\mathrm{osc}}:=\mathrm{ess_{\mu}}\sup
f-\mathrm{ess}_{\mu}\inf f$ and
$\left\|f\right\|_{\infty}:=\mathrm{ess_{\mu}}\sup\left|f\right|$.

  * •

For a finite signed measure $\nu$ on $\left(\mathsf{E},\mathscr{E}\right)$ its
total variation is given by

$\left\|\nu\right\|_{\mathrm{TV}}=\frac{1}{2}\cdot\sup\left\\{\nu\left(f\right):\left\|f\right\|_{\infty}\leqslant
1\right\\}\,,$

which is equal to
$\sup\left\\{\nu\left(f\right):\left\|f\right\|_{\mathrm{osc}}\leqslant
1\right\\}$ when $\nu(\mathsf{E})=0$, e.g. [18, Proposition D.2.4].

  * •

For two probability measures $\mu$ and $\nu$ on
$\left(\mathsf{E},\mathscr{E}\right)$ we let $\mu\otimes\nu\left(A\times
B\right)=\mu\left(A\right)\nu\left(B\right)$ for $A,B\in\mathscr{E}$. For a
Markov kernel $P\left(x,\mathrm{d}y\right)$ on $\mathsf{E}\times\mathscr{E}$,
we write for $\bar{A}\in\mathscr{E}\otimes\mathscr{E}$, the minimal product
$\sigma$-algebra, $\mu\otimes
P\left(\bar{A}\right)=\int_{\bar{A}}\mu\left(\mathrm{d}x\right)P\left(x,\mathrm{d}y\right)$.

  * •

A point mass distribution at $x$ will be denoted by
$\delta_{x}\left(\mathrm{d}y\right)$.

  * •

$\mathrm{Id}:\mathrm{L}^{2}\left(\mu\right)\to\mathrm{L}^{2}\left(\mu\right)$
denotes the identity mapping, $f\mapsto f$. We also use this symbol for the
identity ${\rm Id}\colon\mathsf{X}\rightarrow\mathsf{X}$.

  * •

For a bounded linear operator $T$, we write $T^{*}$ for its adjoint operator
$T^{*}:\mathrm{L}^{2}\left(\mu\right)\to\mathrm{L}^{2}\left(\mu\right)$, which
satisfies $\left\langle f,Tg\right\rangle=\left\langle T^{*}f,g\right\rangle$
for any $f,g\in\mathrm{L}^{2}\left(\mu\right)$. We denote its operator norm by

$\left\|T\right\|_{\mathrm{op}}=\sup\left\\{\left\|Tf\right\|_{2}:f\in\mathrm{L}^{2}\left(\mu\right),\left\|f\right\|_{2}\leqslant
1\right\\}.$

  * •

Given a bounded linear operator
$T:\mathrm{L}^{2}\left(\mu\right)\to\mathrm{L}^{2}\left(\mu\right)$
corresponding to a $\mu$-invariant Markov kernel, we define the Dirichlet form
to be

$\displaystyle\mathcal{E}\left(T,f\right)$
$\displaystyle:=\left\langle\left(\mathrm{Id}-T\right)f,f\right\rangle$
$\displaystyle=\frac{1}{2}\int\mu\left(\mathrm{d}x\right)\cdot
T\left(x,\mathrm{d}y\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|^{2},$

for any $f\in\mathrm{L}^{2}\left(\mu\right)$. More generally, for $p\geqslant
1$ and $f\in\mathrm{L}^{p}\left(\mu\right)$, we also define

$\mathcal{E}_{p}\left(T,f\right):=\frac{1}{2}\cdot\int\mu\left(\mathrm{d}x\right)\cdot
T\left(x,\mathrm{d}y\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|^{p}.$
(28)

  * •

The spectrum of a bounded linear operator $T$ is the set

$\sigma\left(T\right):=\left\\{\lambda\in\mathbb{C}:T-\lambda\cdot{\rm
Id}\text{ is not invertible}\right\\}.$

  * •

For a $\mu$-invariant Markov kernel $T$, we denote by
$\sigma_{0}\left(T\right)$ the spectrum of the restriction of $T$ to
$\mathrm{L}^{2}_{0}\left(\mu\right)$. We define the spectral gap of $T$ to be
$\gamma_{T}:=1-\sup\left|\sigma_{0}\left(T\right)\right|$.

  * •

We say that a Markov kernel $T$ is $\mu$-reversible if $\mu\otimes
T\left(A\times B\right)=\mu\otimes T\left(B\times A\right)$ for all sets
$A,B\in\mathscr{E}$. We say that a $\mu$-reversible kernel $T$ is positive if
for any $f\in\mathrm{L}^{2}\left(\mu\right)$, we have $\left\langle
f,Tf\right\rangle\geqslant 0$.

  * •

If $T$ is a $\mu$-reversible Markov kernel, then
$\sigma_{0}\left(T\right)\subseteq\left[-1,1\right]$ and we may define the
right spectral gap as ${\rm Gap_{{\rm
R}}}\left(T\right):=1-\sup\sigma_{0}\left(T\right)$, which satisfies [18,
Theorem 22.A.19],

${\rm
Gap}_{\mathrm{R}}\left(T\right)=\inf_{g\in\mathrm{L}^{2}_{0}\left(\mu\right),g\neq
0}\frac{\mathcal{E}\left(T,g\right)}{\left\|g\right\|_{2}^{2}}\,.$

  * •

In practice, fluctuations of ergodic averages of
$f\in\mathrm{L}^{2}\left(\mu\right)$ are of interest, and one may consider the
asymptotic variance, given by [39, Proof of Theorem 4]

${\rm var}\left(T,f\right):=\lim_{n\to\infty}n\cdot{\rm
var}\left(\frac{1}{n}\sum_{i=1}^{n}f\left(X_{i}\right)\right),$

where $X_{0}\sim\mu$ and for $i\geqslant 1$, $\mathbb{P}\left(X_{i}\in A\mid
X_{0}=x_{0},\ldots,X_{i-1}=x_{i-1}\right)=T\left(x_{i-1},A\right)$.

  * •

If $T$ is a $\mu$-reversible Markov kernel we may write that [39, Proof of
Theorem 4]

$\displaystyle{\rm var}\left(T,f\right)$ $\displaystyle=\lim_{\lambda\uparrow
1}\left\langle f,\left(\mathrm{Id}-\lambda
T\right)^{-1}\left(\mathrm{Id}+\lambda T\right)f\right\rangle.$

##  Appendix B Dirichlet forms of indicators

######  Lemma 102.

Let
$T\colon\mathrm{L}^{2}\left(\mu\right)\rightarrow\mathrm{L}^{2}\left(\mu\right)$
be a bounded linear operator and $S=\frac{1}{2}\cdot\left(T+T^{*}\right)$ be
its symmetric part. Then

  1. a).

For $p\in\left\\{1,2\right\\}$

$\displaystyle\mathcal{E}_{p}\left(T,f\right)$
$\displaystyle=\mathcal{E}_{p}\left(T^{*},f\right)=\mathcal{E}_{p}\left(S,f\right).$

  2. b).

For $p\in\left\\{1,2\right\\}$ and any $A\in\mathcal{E}$

$\mathcal{E}_{p}\left(T,\mathbf{1}_{A}\right)=\mu\otimes T\big{(}A\times
A^{\complement}\big{)}=\mu\otimes T\big{(}A^{\complement}\times A\big{)}\text{
and
}\mathrm{var}_{\mu}\big{(}\mathbf{1}_{A}\big{)}=\mu\otimes\mu\big{(}A\times
A^{\complement}\big{)}\;.$

###### Proof.

We have for $p\in\left\\{1,2\right\\}$

$\displaystyle\mathcal{E}_{p}\left(T,f\right)$
$\displaystyle=\frac{1}{2}\cdot\int\mu\left(\mathrm{d}x\right)\cdot
T\left(x,\mathrm{d}y\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|^{p}$
$\displaystyle=\frac{1}{2}\cdot\int\mu\left(\mathrm{d}y\right)\cdot
T^{*}\left(y,\mathrm{d}x\right)\cdot\left|f\left(y\right)-f\left(x\right)\right|^{p}$
$\displaystyle=\mathcal{E}_{p}\left(T^{*},f\right),$

and by linearity that
$\mathcal{E}_{p}\left(T,f\right)=\mathcal{E}_{p}\left(w\cdot
T+\left(1-w\right)\cdot T^{*},f\right)$ for any $w\in\left[0,1\right]$. It
follows that
$\mathcal{E}_{p}\left(T,f\right)=\mathcal{E}_{p}\left(S,f\right)$.

Now, for arbitrary $\mu$-invariant $T$, it holds that

$\displaystyle\left(\mu\otimes T\right)\left(A\times A^{\complement}\right)$
$\displaystyle-\left(\mu\otimes T\right)\left(A^{\complement}\times A\right)$
$\displaystyle=\int\mu\left(\mathrm{d}x\right)\cdot
T\left(x,\mathrm{d}y\right)\cdot\left[\mathbf{1}_{A}\left(x\right)\cdot
1_{A^{\complement}}\left(y\right)-\mathbf{1}_{A}\left(y\right)\cdot
1_{A^{\complement}}\left(x\right)\right]$
$\displaystyle=\int\mu\left(\mathrm{d}x\right)\cdot
T\left(x,\mathrm{d}y\right)\cdot\left[\mathbf{1}_{A}\left(x\right)-\mathbf{1}_{A}\left(y\right)\right]$
$\displaystyle=\mu\left(A\right)-\mu
T\left(A\right)=\mu\left(A\right)-\mu\left(A\right)=0,$

i.e. that $\left(\mu\otimes T\right)\left(A\times
A^{\complement}\right)=\left(\mu\otimes T\right)\left(A^{\complement}\times
A\right)$, and that

$\displaystyle\mathcal{E}_{p}\left(T,\mathbf{1}_{A}\right)$
$\displaystyle=\frac{1}{2}\int\left|\mathbf{1}_{A}\left(x\right)-\mathbf{1}_{A}\left(y\right)\right|^{p}\cdot\mu\otimes
T\left({\rm d}x,{\rm d}y\right)$
$\displaystyle=\frac{1}{2}\int\left[\mathbf{1}_{A}\left(x\right)\cdot\mathbf{1}_{A^{\complement}}\left(y\right)+\mathbf{1}_{A^{\complement}}\left(x\right)\cdot\mathbf{1}_{A}\left(y\right)\right]\cdot\mu\otimes
T\left({\rm d}x,{\rm d}y\right)$
$\displaystyle=\frac{1}{2}\left\\{\left(\mu\otimes T\right)\left(A\times
A^{\complement}\right)+\left(\mu\otimes T\right)\left(A^{\complement}\times
A\right)\right\\}.$

Combining these two identities yields the first two equalities. Finally, the
result on the variance follows by considering
$T\left(x,A\right)=\mu\left(A\right)$ for
$\left(x,A\right)\in\mathsf{E}\times\mathscr{E}$ and the classical identity
$\mathrm{var}_{\mu}\big{(}\mathbf{1}_{A}\big{)}=\frac{1}{2}\mathbb{E}_{\mu\otimes\mu}\left[\big{(}\mathbf{1}_{A}\left(X\right)-\mathbf{1}_{A}\left(Y\right)\big{)}^{2}\right]$.
∎

##  Appendix C Improved sharpness of $\mathrm{L}^{1}$_–_ WPI result

In this section we establish that the results of Theorem 35 improves mixing
times estimates compared to those obtained using earlier results in [3,
Theorem 38]. First, since $\mu\otimes P\left(A\times
A^{\complement}\right)=\mu\otimes P\left(A^{\complement}\times A\right)$ for
any $A\in\mathscr{E}$, which does not require reversibility, we have for
$v\in\left[0,\frac{1}{4}\right]$,

$\displaystyle\kappa(v):$ $\displaystyle=\inf\left\\{\frac{\mu\otimes
P\left(A\times
A^{\complement}\right)}{\mu\left(A\right)\mu\left(A^{\complement}\right)}:v\leqslant\mu\left(A\right)\mu\left(A^{\complement}\right)\leqslant\frac{1}{4}\right\\}$
$\displaystyle=\inf\left\\{\frac{\mu\otimes P\left(A\times
A^{\complement}\right)}{\mu\left(A\right)\mu\left(A^{\complement}\right)}:\frac{1-\sqrt{1-4v}}{2}\leqslant\mu\left(A\right)\leqslant\frac{1}{2}\right\\}$
$\displaystyle\geqslant\inf\left\\{\frac{\mu\otimes P\left(A\times
A^{\complement}\right)}{\mu\left(A\right)}:\frac{1-\sqrt{1-4v}}{2}\leqslant\mu\left(A\right)\leqslant\frac{1}{2}\right\\}$
$\displaystyle=\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-4v}}{2}\right).$

It is shown in [3, Theorem 38] that $\kappa\left(v\right)>0$ for
$v\in\left(0,\frac{1}{4}\right)$ implies a $\left(\Psi,\alpha\right)$-WPI with
$\alpha\left(v\right)=2^{4}\cdot\kappa\left(2^{-4}\cdot v\right)^{-2}$. Using
(16) in the proof of Theorem 35, we obtain

$\displaystyle K^{*}\left(v\right)$ $\displaystyle\geqslant\frac{1}{2}\cdot
v\cdot\alpha\left(\frac{v}{2}\right)^{-1}$
$\displaystyle\geqslant\frac{1}{2}\cdot
v\cdot\left(2^{4}\cdot\kappa\left(2^{-5}\cdot v\right)^{-2}\right)^{-1}$
$\displaystyle\geqslant 2^{-5}\cdot
v\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-2^{-3}\cdot
v}}{2}\right)^{2}\cdot$

From Theorem 8, given $f\in\mathrm{L}^{2}_{0}\left(\mu\right)$, for any
$\varepsilon>0$ we have
$\left\|P^{n}f\right\|_{2}^{2}/\left\|f\right\|_{\mathrm{osc}}^{2}\leqslant\varepsilon$
whenever $n\geqslant F\left(\varepsilon\right)$, and we therefore want the
lowest possible upper bound on $F\left(\varepsilon\right)$. In Corollary 36
our new lower bound on $K^{*}(v)$ leads to

$F\left(\varepsilon\right)\leqslant
2^{2}\cdot\int_{\varepsilon}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)^{2}}.$

We show that this is tighter than what is obtained with the old lower bound
above. We have $\frac{1}{2}x\leqslant 1-\sqrt{1-x}\leqslant x$ for
$x\in\left[0,1\right]$, and using that
$u\mapsto\Phi_{P}^{\mathrm{W}}\left(u\right)$ is non-decreasing it holds that

$\Phi_{P}^{\mathrm{W}}\left(2^{-4}\cdot
v\right)\leqslant\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-2^{-3}\cdot
v}}{2}\right)\leqslant\Phi_{P}^{\mathrm{W}}\left(2^{-3}\cdot v\right),$

and therefore

$\displaystyle F\left(\varepsilon\right)$ $\displaystyle\leqslant
2^{2}\cdot\int_{\varepsilon}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)^{2}}$
$\displaystyle\leqslant 2^{2}\cdot\int_{\varepsilon}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-3}\cdot v\right)^{2}}$
$\displaystyle\leqslant 2^{2}\cdot\int_{\varepsilon}^{1/4}\frac{{\rm
d}v}{v\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-2^{-3}\cdot
v}}{2}\right)^{2}}$
$\displaystyle=2^{-3}\cdot\int_{\varepsilon}^{1/4}\frac{\mathrm{d}v}{2^{-5}\cdot
v\cdot\Phi_{P}^{\mathrm{W}}\left(\frac{1-\sqrt{1-2^{-3}\cdot
v}}{2}\right)^{2}},$

i.e. our new result improves estimates of the mixing time by a factor of at
least $8$. If $\Phi$ is moreover convex and satisfies $\Phi\left(0\right)=0$
(both of which are often the case for chains with slower-than-exponential
convergence), we can use the estimate $\Phi_{P}^{\mathrm{W}}\left(2^{-3}\cdot
v\right)\leqslant 2^{-1}\cdot\Phi_{P}^{\mathrm{W}}\left(2^{-2}\cdot v\right)$
to obtain an additional improvement by a factor of $4$.

##  Appendix D Auxiliary results for examples

###  D.1 Examples 92, 93

######  Lemma 103.

Let $U(x)=A\left(\left|x\right|\right)$ for some sufficiently smooth
$A:\mathbb{R}\to\mathbb{R}$. Then it holds that for $x\neq 0$

$\displaystyle\nabla U(x)$
$\displaystyle=A^{{}^{\prime}}\left(\left|x\right|\right)\cdot\frac{x}{\left|x\right|}$
$\displaystyle\nabla^{2}U(x)$
$\displaystyle=A^{{}^{\prime\prime}}\left(\left|x\right|\right)\cdot\left(\frac{x}{\left|x\right|}\right)\cdot\left(\frac{x}{\left|x\right|}\right)^{\top}$
$\displaystyle\quad+\frac{A^{{}^{\prime}}\left(\left|x\right|\right)}{\left|x\right|}\cdot\left\\{I_{d}-\frac{xx^{\top}}{\left|x\right|^{2}}\right\\}.$

For $d\geqslant 2$, it thus holds for $x\neq 0$ that

$\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}=\max\left\\{\left|A^{{}^{\prime\prime}}\left(\left|x\right|\right)\right|,\left|\frac{A^{{}^{\prime}}\left(\left|x\right|\right)}{\left|x\right|}\right|\right\\}.$

###### Proof.

The first two results follow from direct calculation. For the estimate on the
spectral norm, let $v$ be a vector of norm $1$ and write
$v=v_{\perp}+v_{\parallel}$ with $v_{\parallel}=\langle
v,x\rangle\cdot\frac{x}{\left|x\right|}$. Compute directly that
$v^{\top}\nabla^{2}U\left(x\right)v=\left|v_{\parallel}\right|^{2}\cdot
A^{{}^{\prime\prime}}\left(\left|x\right|\right)+\left(1-|v_{\parallel}|^{2}\right)\cdot\frac{A^{{}^{\prime}}\left(\left|x\right|\right)}{\left|x\right|}$;
optimising over $\left|v_{\parallel}\right|^{2}$ yields the claimed result. ∎

######  Lemma 104.

Throughout, let $d\geqslant 2$, $\tau>0$, $\alpha\in\left(0,1\right)$. Then,

  1. a).

For
$U\left(x\right)=\frac{d+\tau}{2}\cdot\log\left(\tau+\left|x\right|^{2}\right)$,
it holds that

$\displaystyle\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}$
$\displaystyle=\frac{\tau+d}{\tau+\left|x\right|^{2}}$
$\displaystyle\sup\left\\{\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}:x\in\mathbb{R}^{d}\right\\}$
$\displaystyle=1+d\cdot\tau^{-1}.$

  2. b).

For $U\left(x\right)=\left(\tau+\left|x\right|^{2}\right)^{\alpha/2}$, it
holds that

$\displaystyle\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}$
$\displaystyle=\alpha\cdot\left(\tau+\left|x\right|^{2}\right)^{\alpha/2-1}$
$\displaystyle\sup\left\\{\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}:x\in\mathbb{R}^{d}\right\\}$
$\displaystyle=\alpha\cdot\tau^{-\left(1-\alpha/2\right)}.$

###### Proof.

Explicit calculations. For a), define
$A\left(r\right)=\frac{d+\tau}{2}\cdot\log\left(\tau+r^{2}\right)$, and
compute that

$\displaystyle A^{{}^{\prime}}\left(r\right)$
$\displaystyle=\left(d+\tau\right)\cdot\frac{r}{\tau+r^{2}}$ $\displaystyle
A^{{}^{\prime\prime}}\left(r\right)$
$\displaystyle=\left(d+\tau\right)\cdot\frac{\tau-r^{2}}{\left(\tau+r^{2}\right)^{2}}.$

By Lemma 103, it follows that if $\left|x\right|=r$, then

$\displaystyle\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}$
$\displaystyle=\max\left\\{\left|A^{{}^{\prime\prime}}\left(r\right)\right|,\left|\frac{A^{{}^{\prime}}\left(r\right)}{r}\right|\right\\}$
$\displaystyle=\max\left\\{\left|\left(d+\tau\right)\cdot\frac{\tau-r^{2}}{\left(\tau+r^{2}\right)^{2}}\right|,\left|\left(d+\tau\right)\cdot\frac{1}{\tau+r^{2}}\right|\right\\}$
$\displaystyle=\frac{\tau+d}{\left(\tau+r^{2}\right)^{2}}\cdot\max\left\\{\left|\tau-r^{2}\right|,\left|\tau+r^{2}\right|\right\\}$
$\displaystyle=\frac{\tau+d}{\tau+r^{2}},$

from which the first result follows. Noting that this expression is
monotonically decreasing in $r$ yields the second result.

For b), define similarly $A\left(r\right)=\left(\tau+r^{2}\right)^{\alpha/2}$,
and compute that

$\displaystyle A^{{}^{\prime}}\left(r\right)$ $\displaystyle=\alpha\cdot
r\cdot\left(\tau+r^{2}\right)^{\alpha/2-1}$ $\displaystyle
A^{{}^{\prime\prime}}\left(r\right)$
$\displaystyle=\alpha\cdot\left(\tau+r^{2}\right)^{\alpha/2-1}-\alpha\cdot\left(2-\alpha\right)\cdot
r^{2}\cdot\left(\tau+r^{2}\right)^{\alpha/2-2}$
$\displaystyle=\alpha\cdot\left(\tau+r^{2}\right)^{\alpha/2-2}\cdot\left\\{\tau-\left(1-\alpha\right)\cdot
r^{2}\right\\}$

whence if $\left|x\right|=r$, then

$\displaystyle\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}$
$\displaystyle=\max$
$\displaystyle\left\\{\left|A^{{}^{\prime\prime}}\left(r\right)\right|,\left|\frac{A^{{}^{\prime}}\left(r\right)}{r}\right|\right\\}$
$\displaystyle=\max$
$\displaystyle\left\\{\left|\alpha\cdot\left(\tau+r^{2}\right)^{\alpha/2-2}\cdot\left\\{\tau-\left(1-\alpha\right)\cdot
r^{2}\right\\}\right|,\left|\alpha\cdot\left(\tau+r^{2}\right)^{\alpha/2-1}\right|\right\\}$
$\displaystyle=\alpha\cdot$
$\displaystyle\left(\tau+r^{2}\right)^{-\left(2-\alpha/2\right)}\cdot\max\left\\{\left|\left\\{\tau-\left(1-\alpha\right)\cdot
r^{2}\right\\}\right|,\left|\tau+r^{2}\right|\right\\}$
$\displaystyle=\alpha\cdot$
$\displaystyle\left(\tau+r^{2}\right)^{-\left(1-\alpha/2\right)},$

yielding the first result; monotonicity again yields the second. ∎

######  Lemma 105.

Let $U:\mathbb{R}^{d}\to\mathbb{R}$ be a twice-differentiable mapping such
that
$\sup\left\\{\left\|\nabla^{2}U\left(x\right)\right\|_{\mathrm{spec}}:x\in\mathbb{R}^{d}\right\\}\leqslant
L$ for some $L\in[0,\infty)$. Then $U$ is $L$-smooth.

###### Proof.

Let $x,h\in\mathsf{E}$, and write

$\displaystyle U\left(x+h\right)-U\left(x\right)-\left\langle\nabla
U\left(x\right),h\right\rangle$ $\displaystyle=\int_{0}^{1}\left\langle\nabla
U\left(x+t\cdot h\right)-\nabla U\left(x\right),h\right\rangle\,\mathrm{d}t$
$\displaystyle=\int_{0}^{1}\int_{0}^{t}h^{\top}\nabla^{2}U\left(x+s\cdot
h\right)h\,\mathrm{d}s\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\int_{0}^{t}\left\|\nabla^{2}U\left(x+s\cdot
h\right)\right\|_{\mathrm{spec}}\cdot\left|h\right|^{2}\,\mathrm{d}s\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\int_{0}^{t}L\cdot\left|h\right|^{2}\,\mathrm{d}s\,\mathrm{d}t$
$\displaystyle=\frac{L}{2}\cdot\left|h\right|^{2}.$

∎

###  D.2 Example 82

A ${\rm Pareto}\left(\alpha,x_{m}\right)$ with $x_{m}=1-\alpha^{-1}$ and
$\alpha>1$ for $Q_{*}$ leads to a cumulative distribution function for
$\tilde{\pi}_{*}$

$\tilde{\pi}_{*}\left(W\leqslant s\right)=\begin{cases}0&s\leqslant x_{m}\\\
1-\left(x_{m}/s\right)^{\alpha-1}&s\geqslant x_{m},\end{cases}$

because

$\displaystyle\alpha x_{m}^{\alpha}\int_{x_{m}}^{s}x^{-(1+\alpha)}x\,{\rm d}x$
$\displaystyle=\frac{\alpha
x_{m}^{\alpha}}{\alpha-1}\cdot\left[x_{m}^{-(\alpha-1)}-s^{-(\alpha-1)}\right]$
$\displaystyle=\frac{\alpha
x_{m}}{\alpha-1}\cdot\left[1-\left(x_{m}/s\right)^{\alpha-1}\right]$
$\displaystyle=1-\left(x_{m}/s\right)^{\alpha-1},$

using basic properties of the Pareto distribution. The expression for
$\psi_{*}^{-1}$ therefore follows.

## References

  * [1] Christophe Andrieu, Anthony Lee, Sam Power, and Andi Q. Wang.  Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC.  The Annals of Statistics, 50(6):3592 – 3618, 2022.
  * [2] Christophe Andrieu, Anthony Lee, Sam Power, and Andi Q. Wang.  Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles, 2022.
  * [3] Christophe Andrieu, Anthony Lee, Sam Power, and Andi Q. Wang.  Poincaré inequalities for Markov chains: a meeting with Cheeger, Lyapunov and Metropolis.  Technical report: https://arxiv.org/abs/2208.05239, 2022.
  * [4] Christophe Andrieu and Gareth O. Roberts.  The pseudo-marginal approach for efficient Monte Carlo computations.  The Annals of Statistics, 37(2):697 – 725, 2009.
  * [5] Christophe Andrieu and Matti Vihola.  Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms.  The Annals of Applied Probability, 25(2):1030–1077, 2015.
  * [6] Christophe Andrieu and Matti Vihola.  Establishing some order amongst exact approximations of MCMCs.  The Annals of Applied Probability, 26(5):2661–2696, 10 2016.
  * [7] Dominique Bakry, Ivan Gentil, Michel Ledoux, et al.  Analysis and geometry of Markov diffusion operators, volume 103\.  Springer, 2014.
  * [8] Peter H. Baxendale.  Renewal theory and computable convergence rates for geometrically ergodic Markov chains.  The Annals of Applied Probability, 15(1B):700–738, 2005.
  * [9] Sergey Bobkov.  Large deviations and isoperimetry over convex probability measures with heavy tails.  Electronic Journal of Probability, 12(none):1072 – 1100, 2007.
  * [10] Sergey Bobkov and Boguslaw Zegarlinski.  Distributions with slow tails and ergodicity of markov semigroups in infinite dimensions.  Around the Research of Vladimir Maz’ya I: Function Spaces, pages 13–79, 2009.
  * [11] Sergey G Bobkov and Michel Ledoux.  Weighted poincaré-type inequalities for cauchy and other convex measures.  2009\.
  * [12] Serguei Germanovich Bobkov and Christian Houdré.  Some connections between isoperimetric and Sobolev-type inequalities, volume 616.  American Mathematical Soc., 1997.
  * [13] Patrick Cattiaux, Djalil Chafai, and Arnaud Guillin.  Central limit theorems for additive functionals of ergodic Markov diffusions processes.  ALEA, Lat. Am. J. Probab. Math. Stat, 9(2):337–382, 2012.
  * [14] Patrick Cattiaux, Nathael Gozlan, Arnaud Guillin, and Cyril Roberto.  Functional inequalities for heavy tailed distributions and application to isoperimetry.  Electronic Journal of Probability, 15:346–385, 2010.
  * [15] Patrick Cattiaux, Arnaud Guillin, and Cyril Roberto.  Poincaré inequality and the $L^{p}$ convergence of semi-groups.  Electronic Communications in Probability, 15(none):270 – 280, 2010\.
  * [16] Yuansi Chen, Raaz Dwivedi, Martin J. Wainwright, and Bin Yu.  Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients.  Journal of Machine Learning Research, 21(92):1–72, 2020.
  * [17] Persi Diaconis and Laurent Saloff-Coste.  Nash inequalities for finite Markov chains.  Journal of Theoretical Probability, 9(2):459–510, 1996.
  * [18] Randal Douc, Eric Moulines, Pierre Priouret, and Philippe Soulier.  Markov Chains.  Springer, 2018.
  * [19] Raaz Dwivedi, Yuansi Chen, Martin J. Wainwright, and Bin Yu.  Log-concave sampling: Metropolis–Hastings algorithms are fast.  Journal of Machine Learning Research, 20(183):1–42, 2019.
  * [20] Fuzhou Gong and Liming Wu.  Spectral gap of positive operators and applications.  Journal de Mathématiques Pures et Appliquées, 85(2):151–191, 2006.
  * [21] Søren F. Jarner and Gareth O. Roberts.  Polynomial convergence rates of Markov chains.  The Annals of Applied Probability, 12(1):224–247, 2002.
  * [22] Søren F. Jarner and Gareth O. Roberts.  Convergence of heavy-tailed Monte Carlo Markov chain algorithms.  Scandinavian Journal of Statistics, 34(4):781–815, 2007.
  * [23] Søren F. Jarner and Richard L. Tweedie.  Necessary conditions for geometric and polynomial ergodicity of random-walk-type.  Bernoulli, 9(4):559 – 578, 2003.
  * [24] Gregory F. Lawler and Alan D. Sokal.  Bounds on the $L^{2}$ spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality.  Transactions of the American Mathematical Society, 309(2):557–580, 1988.
  * [25] Anthony Lee and Krzysztof Łatuszyński.  Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation.  Biometrika, 101(3):655–671, 08 2014.
  * [26] Thomas M Liggett.  $L_{2}$ rates of convergence for attractive reversible nearest particle systems: the critical case.  The Annals of Probability, 19(3):935–959, 1991.
  * [27] Michael Maxwell and Michael Woodroofe.  Central limit theorems for additive functionals of Markov chains.  The Annals of Probability, pages 713–724, 2000.
  * [28] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller.  Equation of State Calculations by Fast Computing Machines.  The Journal of Chemical Physics, 21(6):1087–1092, 1953.
  * [29] S.P. Meyn and R.L. Tweedie.  Markov chains and stochastic stability.  Cambridge University Press, 2 edition, 2009.
  * [30] Ravi Montenegro and Prasad Tetali.  Mathematical aspects of mixing times in Markov chains.  Foundations and Trends® in Theoretical Computer Science, 1(3):237–354, 2006.
  * [31] Alireza Mousavi-Hosseini, Tyler Farghly, Ye He, Krishnakumar Balasubramanian, and Murat A Erdogdu.  Towards a complete analysis of langevin monte carlo: Beyond poincaré inequality.  3 2023.
  * [32] Gareth O. Roberts, Andrew Gelman, and Walter R. Gilks.  Weak convergence and optimal scaling of random walk Metropolis algorithms.  The Annals of Applied Probability, 7(1):110–120, 1997.
  * [33] Gareth O. Roberts and Jeffrey S. Rosenthal.  Optimal scaling for various Metropolis–Hastings algorithms.  Statistical Science, 16(4):351–367, 2001.
  * [34] Gareth O. Roberts and Jeffrey S. Rosenthal.  Variance bounding Markov chains.  Ann. Appl. Probab., (3):1201–1214, 2008.
  * [35] Gareth O. Roberts and Jeffrey S. Rosenthal.  Complexity bounds for markov chain monte carlo algorithms via diffusion limits.  Journal of Applied Probability, 53(2):410–420, 2016.
  * [36] Gareth O. Roberts and Richard L. Tweedie.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms.  Biometrika, 83(1):95–110, 1996.
  * [37] Gareth O Roberts and Richard L Tweedie.  Geometric $l^{2}$ and $l^{1}$ convergence are equivalent for reversible markov chains.  Journal of Applied Probability, 38(A):37–41, 2001.
  * [38] Michael Röckner and Feng-Yu Wang.  Weak Poincaré inequalities and L2 convergence rates of Markov semigroups.  Journal of Functional Analysis, 185:564–603, 2001.
  * [39] Luke Tierney.  A note on Metropolis–Hastings kernels for general state spaces.  The Annals of Applied Probability, 8(1):1–9, 1998.
  * [40] Feng Yu Wang.  Criteria of spectral gap for Markov operators.  Journal of Functional Analysis, 266(4):2137–2152, 2014.
  * [41] Pierre-André Zitt.  Annealing diffusions in a potential function with a slow growth.  Stochastic Processes and their Applications, 118(1):76–119, 2008\.