# Edgeworth expansions for independent bounded integer valued random
variables.

Dmitry Dolgopyat and Yeor Hafouta

###### Abstract.

We obtain asymptotic expansions for local probabilities of partial sums for
uniformly bounded independent but not necessarily identically distributed
integer-valued random variables. The expansions involve products of
polynomials and trigonometric polynomials. Our results do not require any
additional assumptions. As an application of our expansions we find necessary
and sufficient conditions for the classical Edgeworth expansion. It turns out
that there are two possible obstructions for the validity of the Edgeworth
expansion of order $r$. First, the distance between the distribution of the
underlying partial sums modulo some $h\in{\mathbb{N}}$ and the uniform
distribution could fail to be $o(\sigma_{N}^{1-r})$, where $\sigma_{N}$ is the
standard deviation of the partial sum. Second, this distribution could have
the required closeness but this closeness is unstable, in the sense that it
could be destroyed by removing finitely many terms. In the first case, the
expansion of order $r$ fails. In the second case it may or may not hold
depending on the behavior of the derivatives of the characteristic functions
of the summands whose removal causes the break-up of the uniform distribution.
We also show that a quantitative version of the classical Prokhorov condition
(for the strong local central limit theorem) is sufficient for Edgeworth
expansions, and moreover this condition is, in some sense, optimal.

###### Contents

  1. 1 Introduction.
  2. 2 Main result
  3. 3 Edgeworth expansions under quantitative Prokhorov condition.
    1. 3.1 Characterstic function near 0.
    2. 3.2  Non resonant intervals.
    3. 3.3 Prokhorov estimates
    4. 3.4  Proof of Theorem 1.4
  4. 4 Generalized Edgeworth expansions: Proof of Theorem Theorem 2.1
    1. 4.1 Contributions of resonant intervals.
    2. 4.2 Completing the proof
    3. 4.3 A summary
  5. 5 Uniqueness of trigonometric expansions.
  6. 6 First order expansions
  7. 7 Characterizations of Edgeworth expansions of all orders.
    1. 7.1 Derivatives of the non-perturbative factor.
    2. 7.2 Proof of Lemma 7.2.
    3. 7.3 Proof of Theorem 1.5
  8. 8 Edgeworth expansions and uniform distribution.
    1. 8.1 Proof of Theorem 1.6
    2. 8.2 Proof of Theorem 1.8
  9. 9 Second order expansions
  10. 10 Examples.
  11. 11 Extension for uniformly bounded integer-valued triangular arrays

##  1\. Introduction.

Let $X_{1},X_{2},...$ be a uniformly bounded sequence of independent integer-
valued random variables. Set $S_{N}=X_{1}+X_{2}+...+X_{N}$,
$V_{N}=V(S_{N})=\text{Var}(S_{N})$ and ${\sigma}_{N}=\sqrt{V_{N}}$. Assume
also that $V_{N}\to\infty$ as $N\to\infty$. Then the central limit theorem
(CLT) holds true, namely the distribution of
$(S_{N}-{\mathbb{E}}(S_{N}))/{\sigma}_{N}$ converges to the standard normal
distribution as $N\to\infty$.

Recall that the local central limit theorem (LLT) states that, uniformly in
$k$ we have

${\mathbb{P}}(S_{N}=k)=\frac{1}{\sqrt{2\pi}{\sigma}_{N}}e^{-\left(k-{\mathbb{E}}(S_{N})\right)^{2}/2V_{N}}+o({\sigma}_{N}^{-1}).$

This theorem is also a classical result, and it has origins in De Moivre-
Laplace theorem. The stable local central limit theorem (SLLT) states that the
LLT holds true for any integer-valued square integrable independent sequence
$X_{1}^{\prime},X_{2}^{\prime},...$ which differs from $X_{1},X_{2},...$ by a
finite number of elements. We recall a classical result due to Prokhorov.

######  Theorem 1.1.

[32] The SLLT holds iff for each integer $h>1$,

(1.1) $\sum_{n}{\mathbb{P}}(X_{n}\neq m_{n}\text{ mod }h)=\infty$

where $m_{n}=m_{n}(h)$ is the most likely residue of $X_{n}$ modulo $h$.

We refer the readers’ to [33, 36] for extensions of this result to the case
when $X_{n}$’s are not necessarily bounded (for instance, the result holds
true when $\displaystyle\sup_{n}\|X_{n}\|_{L^{3}}<\infty$). Related results
for local convergence to more general limit laws are discussed in [6, 25].

The above result provides a necessary and sufficient condition for the SLLT.
It turns out that the difference between LLT and SLLT is not that big.

######  Proposition 1.2.

Suppose $S_{N}$ obeys LLT. Then for each integer $h\geq 2$ at least one of the
following conditions occur:

either (a) $\displaystyle\sum_{n}{\mathbb{P}}(X_{n}\neq m_{n}(h)\text{ mod
}h)=\infty$.

or (b) $\exists j_{1},j_{2},\dots,j_{k}$ with $k<h$ such that
$\displaystyle\sum_{s=1}^{k}X_{j_{s}}$ mod $h$ is uniformly distributed. In
that case for all $N\geq\max(j_{1},\dots,j_{k})$ we have that $S_{N}$ mod $h$
is uniformly distributed.

Since we could not find this result in the literature we include the proof in
Section 6.

Next, we provide necessary and sufficient conditions for the regular LLT. We
need an additional notation. Let $\displaystyle
K=\sup_{n}\|X_{n}\|_{L^{\infty}}$. Call $t$ resonant if $t=\frac{2\pi l}{m}$
with $0<m\leq 2K$ and $0\leq l<m.$

######  Theorem 1.3.

The following conditions are equivalent:

(a) $S_{N}$ satisfies LLT;

(b) For each $\xi\in{\mathbb{R}}\setminus{\mathbb{Z}}$,
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left(e^{2\pi i\xi
S_{N}}\right)=0$;

(c) For each non-zero resonant point $\xi$,
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left(e^{2\pi i\xi
S_{N}}\right)=0$;

(d) For each integer $h$ the distribution of $S_{N}$ mod $h$ converges to
uniform.

The proof of this result is also given in Section 6. We refer the readers to
[7, 10] for related results in more general settings.

The local limit theorem deals with approximation of $P(S_{N}=k)$ up to an
error term of order $o({\sigma}_{N}^{-1})$. Given $r\geq 1$, the Edgeworth
expansion of order $r$ holds true if there are polynomials $P_{b,N}$, whose
coefficients are uniformly bounded in $N$ and their degrees do no depend on
$N,$ so that uniformly in $k\in{\mathbb{Z}}$ we have that

(1.2)
${\mathbb{P}}(S_{N}=k)=\sum_{b=1}^{r}\frac{P_{b,N}(k_{N})}{\sigma_{N}^{b}}\mathfrak{g}(k_{N})+o(\sigma_{N}^{-r})$

where $k_{N}=\left(k-{\mathbb{E}}(S_{N})\right)/{\sigma}_{N}$ and
$\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}}e^{-u^{2}/2}.$ In Section 5 we will
show, in particular, that Edgeworth expansions of any order $r$ are unique up
to terms of order $o({\sigma}_{N}^{-r}),$ and so the case $r=1$ coincides with
the LLT. Edgeworth expansions for discrete (lattice-valued) random variables
have been studied in literature for iid random variables [18, Theorem 4.5.4]
[30, Chapter VII], (see also [12, Theorem 5]), homogeneous Markov chains [28,
Theorems 2-4], decomposable statistics [22], or dynamical systems [15] with
good spectral properties such as expanding maps. Papers [2, 16] discuss the
rate of convergence in the LLT. Results for non-lattice variables were
obtained in [13, 1, 5, 3] (which considered random vectors) and [15] (see also
[17] for corresponding results for random expanding dynamical systems).

In this paper we obtain analogues of Theorems 1.1 and 1.3 for higher order
Edgeworth expansions for independent but not identically distributed integer-
valued uniformly bounded random variables. We begin with the following result.

######  Theorem 1.4.

Let $\displaystyle K=\sup_{j}\|X_{j}\|_{L^{\infty}}.$ For each
$r\in{\mathbb{N}}$ there is a constant $R\\!\\!=\\!\\!R(r,K)$ such that the
Edgeworth expansion of order $r$ holds if

$M_{N}:=\min_{2\leq h\leq 2K}\sum_{n=1}^{N}{\mathbb{P}}(X_{n}\neq
m_{n}(h)\text{ mod }h)\geq R\ln V_{N}.$

In particular, $S_{N}$ obeys Edgeworth expansions of all orders if

$\lim_{N\to\infty}\frac{M_{N}}{\ln V_{N}}=\infty.$

The number $R(r,K)$ can be chosen according to Remark 3.6. This theorem is a
quantitative version of Prokhorov’s Theorem 1.1. We observe that logarithmic
in $V_{N}$ growth of various non-perioidicity characteristics of individual
summands are often used in the theory of local limit theorems (see e.g. [20,
21, 23]). We will see from the examples of Section 10 that this result is
close to optimal. However, to justify the optimality we need to understand the
conditions necessary for the validity of the Edgeworth expansion.

######  Theorem 1.5.

For any $r\geq 1$, the Edgeworth expansion of order $r$ holds if and only if
for any nonzero resonant point $t$ and $0\leq\ell<r$ we have

$\bar{\Phi}_{N}^{(\ell)}(t)=o\left({\sigma}_{N}^{\ell+1-r}\right).$

where $\bar{\Phi}_{N}(x)={\mathbb{E}}[e^{ix(S_{N}-{\mathbb{E}}(S_{N}))}]$ and
$\bar{\Phi}_{N}^{(\ell)}(\cdot)$ is its $\ell$-th derivative.

This results generalizes Theorem 1.3, however in contrast with that theorem,
in the case $r>1$ we also need to take into account the behavior of the
derivatives of the characteristic function at nonzero resonant points. The
values of the characteristic function at the resonant points $2\pi l/m$ have
clear probabilistic meaning. Namely, they control the rate equidistribution
modulo $m$ (see part (d) of Theorem 1.3 or Lemma 6.2). Unfortunately, the
probabilistic meaning of the derivatives is less clear, so it is desirable to
characterize the validity of the Edgeworth expansions of orders higher than 1
without considering the derivatives. Example 10.2 shows that this is
impossible without additional assumptions. Some of the reasonable additional
conditions are presented below.

We start with the expansion of order 2.

######  Theorem 1.6.

Suppose $S_{N}$ obeys the SLLT. Then the following are equivalent:

(a) Edgeworth expansion of order 2 holds;

(b) $|\Phi_{N}(t)|=o(\sigma_{N}^{-1})$ for each nonzero resonant point $t$;

(c) For each $h\leq 2K$ the distribution of $S_{N}$ mod $h$ is
$o(\sigma_{N}^{-1})$ close to uniform.

Corollary 7.3 provides an extension of Theorem 1.6 for expansions of an
arbitrary order $r$ under an additional assumption that
$\displaystyle\varphi:=\min_{t\in{\mathcal{R}}}\inf_{n}|\phi_{n}(t)|>0$, where
${\mathcal{R}}$ is the set of all nonzero resonant points. The latter
condition implies in particular, that for each $\ell$ there is a uniform lower
bound on the distance between the distribution of
$X_{n_{1}}+X_{n_{2}}+\dots+X_{n_{\ell}}$ mod $m$ and the uniform distribution,
when $\\{n_{1},n_{2},\dots,n_{\ell}\\}\in{\mathbb{N}}^{\ell}$ and $m\geq 2$.

Next we discuss an analogue of Theorem 1.1 for expansions of order higher than
2. It requires a stronger condition which uses an additional notation. Given
$j_{1},j_{2},\dots,j_{s}$ with $j_{l}\in[1,N]$ we write

$S_{N;j_{1},j_{2},\dots,j_{s}}=S_{N}-\sum_{l=1}^{s}X_{j_{l}}.$

Thus $S_{N;j_{1},j_{2},\dots,j_{s}}$ is a partial sum of our sequence with $s$
terms removed. We will say that $\\{X_{n}\\}$ admits an Edgeworth expansion of
order $r$ in a superstable way (which will be denoted by $\\{X_{n}\\}\in
EeSs(r)$) if for each ${\bar{s}}$ and each sequence
$j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$ with $s_{N}\leq{\bar{s}}$ there are
polynomials $P_{b,N}$ whose coefficients are $O(1)$ in $N$ and their degrees
do not depend on $N$ so that uniformly in $k\in{\mathbb{Z}}$ we have that

(1.3)
${\mathbb{P}}(S_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}=k)=\sum_{b=1}^{r}\frac{P_{b,N}(k_{N})}{\sigma_{N}^{b}}\mathfrak{g}(k_{N})+o(\sigma_{N}^{-r})$

and the estimates in $O(1)$ and $o(\sigma_{N}^{-r})$ are uniform in the choice
of the tuples $j_{1}^{N},\dots,j_{s_{N}}^{N}.$ That is, by removing a finite
number of terms we can not destroy the validity of the Edgeworth expansion
(even though the coefficients of the underlying polynomials will of course
depend on the choice of the removed terms). Let
$\Phi_{N;j_{1},j_{2},\dots,j_{s}}(t)$ be the characteristic function of
$S_{N;j_{1},j_{2},\dots,j_{s}}.$

######  Remark 1.7.

We note that in contrast with SLLT, in the definition of the superstrong
Edgeworth expansion one is only allowed to remove old terms, but not to add
new ones. This difference in the definition is not essential, since adding
terms with sufficiently many moments (in particular, adding bounded terms)
does not destroy the validity of the Edgeworth expansion. See the proof of
Theorem 1.8 (i) or the second part of Example 10.1, starting with equation
(10.2), for details.

######  Theorem 1.8.

(1) $S_{N}\in EeSs(1)$ (that is, $S_{N}$ satisfies the LLT in a superstable
way) if and if it satisfies the SLLT.

(2) For arbitrary $r\geq 1$ the following conditions are equivalent:

(a) $\\{X_{n}\\}\in EeSs(r)$;

(b) For each $j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$ and each nonzero
resonant point $t$ we have
$\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma_{N}^{1-r});$

(c) For each $j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$, and each $h\leq 2K$
the distribution of $S_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}$ mod $h$ is
$o(\sigma_{N}^{1-r})$ close to uniform.

To prove the above results we will show that for any order $r$, we can always
approximate $\mathbb{P}(S_{N}=k)$ up to an error $o(\sigma_{N}^{-r})$ provided
that instead of polynomials we use products of regular and the trigonometric
polynomials. Those products allow us to take into account possible oscillatory
behavior of $P(S_{N}=k)$ when $k$ belongs to different residues mod $h$, where
$h$ is denominator of a resonant frequency. When $M_{N}\geq RV_{N}$ for $R$
large enough, the new expansion coincides with the usual Edgeworth expansions.
We thus derive that the condition $M_{N}\geq R\ln V_{N}$ is in a certain sense
optimal.

##  2\. Main result

Let $X_{1},X_{2},...$ be a sequence of independent integer-valued random
variables. For each $N\in{\mathbb{N}}$ we set $\displaystyle
S_{N}=\sum_{n=1}^{N}X_{n}$ and $V_{N}=\text{Var}(S_{N})$. We assume in this
paper that $\displaystyle\lim_{N\to\infty}V_{N}=\infty$ and that there is a
constant $K$ such that

$\sup_{n}\|X_{n}\|_{L^{\infty}}\leq K.$

Denote $\sigma_{N}=\sqrt{V_{N}}.$ For each positive integer $m$, let
$q_{n}(m)$ be the second largest among

$\displaystyle\sum_{l\equiv j\text{ mod
}m}{\mathbb{P}}(X_{n}=l)={\mathbb{P}}(X_{n}\equiv j\text{ mod
}m),\,j=1,2,...,m$

and $j_{n}(m)$ be the corresponding residue class. Set

$M_{N}(m)=\sum_{n=1}^{N}q_{n}(m)\quad\text{and}\quad M_{N}=\min_{m}M_{N}(m).$

######  Theorem 2.1.

There $\exists J=J(K)<\infty$ and polynomials $P_{a,b,N}$, where $a\in
0,\dots,J-1,$ $b\in\mathbb{N}$, with degrees depending only on $b$ but not on
$a,K$ or on any other characteristic of $\\{X_{n}\\}$, such that the
coefficients of $P_{a,b,N}$ are uniformly bounded in $N$, and, for any $r\geq
1$ uniformly in $k\in{\mathbb{Z}}$ we have

${\mathbb{P}}(S_{N}=k)-\sum_{a=0}^{J-1}\sum_{b=1}^{r}\frac{P_{a,b,N}((k-a_{N})/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}((k-a_{N})/\sigma_{N})e^{2\pi
iak/J}=o(\sigma_{N}^{-r})$

where $a_{N}={\mathbb{E}}(S_{N})$ and
$\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}}e^{-u^{2}/2}.$

Moreover, $P_{0,1,N}\equiv 1$, and given $K,r$, there exists $R=R(K,r)$ such
that if $M_{N}\geq R\ln V_{N}$ then we can choose $P_{a,b,N}=0$ for $a\neq 0.$

We refer the readers to (6.1) for more details on these expansions in the case
$r=1$, and to Section 6 for a discussion about the relations with local limit
theorems. The resulting expansions in the case $r=2$ are given in (9.1). We
note that the constants $J(K)$ and $R(K,r)$ can be recovered from the proof of
Theorem 2.1.

######  Remark 2.2.

Since the coefficients of the polynomials $P_{a,b,N}$ are uniformly bounded,
the terms corresponding to $b=r+1$ are of order $O({\sigma}_{N}^{-(r+1)})$
uniformly in $k$. Therefore, in the $r$-th order expansion we actually get
that the error term is $O({\sigma}_{N}^{-(r+1)})$.

######  Remark 2.3.

In fact, the coefficients of the polynomials $P_{a,b,N}$ for $a>0$ are bounded
by a constant times $(1+M_{N}^{q})e^{-c_{0}M_{N}}$, where $c_{0}>0$ depends
only on $K$ and $q\geq 0$ depends only on $r$ and $K$. Therefore, these
coefficient are small when $M_{N}$ is large. When $M_{N}\geq R(r,K)\ln V_{N}$
these coefficients become of order $o({\sigma}^{-r}_{N}).$ Therefore, they
only contribute to the error term, and so we can replace them by $0$, as
stated in Theorem 2.1.

######  Remark 2.4.

As in the derivation of the classical Edgeworth expansion, the main idea of
the proof of Theorem 2.1 is the stationary phase analysis of the
characteristic function. However, in contrast with the iid case there may be
resonances other than 0 which contribute to the oscillatory terms in the
expansion. Another interesting case where the classical Edgeworth analysis
fails is the case of iid terms where the summands are non-arithmetic but take
only finitely many values. It is shown in [8] that in that case, the leading
correction to the Edgeworth expansion also comes from resonances. However, in
the case studied in [8] the geometry of resonances is more complicated, so in
contrast to our Theorem 2.1, [8] does not get the expansion of all orders.

##  3\. Edgeworth expansions under quantitative Prokhorov condition.

In this section we prove Theorem 1.4. In the course of the proof we obtain the
estimates of the characteristic function on intervals not containing resonant
points which will also play an important role in the proof of Theorem 2.1. The
proof of Theorem 2.1 will be completed in Section 4 where we analyze
additional contribution coming from nonzero resonant points which appear in
the case $M_{N}\leq R\ln{\sigma}_{N}.$ Those contributions constitute the
source of the trigonometric polynomials in the generalized Edgeworth
expansions.

###  3.1. Characterstic function near 0.

Here we recall some facts about the behavior of the characteristic function
near 0, which will be useful in the proofs of Theorems 1.4 and 2.1. The first
result holds for general uniformly bounded sequences $\\{X_{n}\\}$ (which are
not necessarily integer-valued).

######  Proposition 3.1.

Suppose that $\displaystyle\lim_{N\to\infty}{\sigma}_{N}=\infty$, where
${\sigma}_{N}\\!=\\!\sqrt{V_{N}}=\sqrt{V(S_{N})}$. Then for $k=1,2,3,...$
there exists a sequence of polynomials $(A_{k,N})_{N}$ whose degree $d_{k}$
depends only on $k$ so that for any $r\geq 1$ there is ${\delta}_{r}>0$ such
that for all $N\geq 1$ and
$t\in[-{\delta}_{r}{\sigma}_{N},{\delta}_{r}{\sigma}_{N}]$,

(3.1)
${\mathbb{E}}\left(e^{it(S_{N}-{\mathbb{E}}(S_{N}))/{\sigma}_{N}}\right)=e^{-t^{2}/2}\left(1+\sum_{k=1}^{r}\frac{A_{k,N}(t)}{{\sigma}_{N}^{k}}+\frac{t^{r+1}}{{\sigma}_{N}^{r+1}}O(1)\right).$

Moreover, the coefficients of $A_{k,N}$ are algebraic combinations of moments
of the $X_{m}$’s and they are uniformly bounded in $N$. Furthermore

(3.2) $A_{1,N}(t)=-\frac{i}{6}{\gamma}_{N}t^{3}\,\,\text{ and
}\,\,A_{2,N}(t)=\Lambda_{4}(\bar{S}_{N}){\sigma}_{N}^{-2}\frac{t^{4}}{4!}-\frac{1}{36}{\gamma}_{N}^{2}t^{6}$

where $\bar{S}_{N}=S_{N}-{\mathbb{E}}(S_{N})$,
${\gamma}_{N}={\mathbb{E}}[(\bar{S}_{N})^{3}]/{\sigma}_{N}^{2}$ and
$\Lambda_{4}(\bar{S}_{N})$ is the fourth comulant of $\bar{S}_{N}$.

The proof is quite standard, so we just sketch the argument. The idea is to
fix some $B_{2}>B_{1}>0$, and to partition $\\{1,...,N\\}$ into intervals
$I_{1},...,I_{m_{N}}$ so that $\displaystyle
B_{1}\leq\text{Var}(S_{I_{l}})\leq B_{2}$ where for each $l$ we set
$\displaystyle S_{I_{l}}=\sum_{j\in I_{l}}X_{i}$. It is clear that
$m_{N}/{\sigma}_{N}^{2}$ is bounded away from $0$ and $\infty$ uniformly in
$N$. Recall next that there are constants $C_{p}$, $p\geq 2$ so that for any
$n\geq 1$ and $m\geq 0$ we have

(3.3)
$\left\|\sum_{j=n}^{n+m}\big{(}X_{j}-{\mathbb{E}}(X_{j})\big{)}\right\|_{L^{p}}\leq
C_{p}\left(1+\left\|\sum_{j=n}^{n+m}\big{(}X_{j}-{\mathbb{E}}(X_{j})\big{)}\right\|_{L^{2}}\right).$

This is a consequence of the multinomial theorem and some elementary
estimates, and we refer the readers to either Lemma 2.7 in [10], or Theorem
6.17 in [24] for such a result in a much more general settings. Using the
latter estimates we get that the $L^{p}$-norms of $S_{I_{l}}$ are uniformly
bounded in $l$. This reduces the problem to the case when the variance of
$X_{n}$ is uniformly bounded from below, and all the moments of
$X_{n}-{\mathbb{E}}(X_{n})$ are uniformly bounded. In this case, the
proposition follows by considering the Taylor expansion of the function
$\ln{\mathbb{E}}\big{(}e^{it(S_{N}-{\mathbb{E}}(S_{N}))/{\sigma}_{N}}\big{)}+\frac{1}{2}t^{2}$,
see [13, §XVI.6].

######  Proposition 3.2.

Given a square integrable random variable $X$, let
$\bar{X}=X-{\mathbb{E}}(X).$ Then for each $h\in\mathbb{R}$ we have

$\left|{\mathbb{E}}(e^{ih\bar{X}})-1\right|\leq\frac{1}{2}h^{2}V(X).$

###### Proof.

Set $\varphi(h)={\mathbb{E}}(e^{ih\bar{X}})$. Then by the integral form of the
second order Taylor reminder we have

$|\varphi(h)-\varphi(0)-h\varphi^{\prime}(0)|=|\varphi(h)-\varphi(0)|=\left|\int_{0}^{h}(t-h)\varphi^{\prime\prime}(t)dt\right|$

$\hskip 133.72786pt\displaystyle\leq
V(X)\int_{0}^{|h|}(|h|-t)dt=\frac{1}{2}h^{2}V(X).$ ∎

###  3.2.  Non resonant intervals.

As in almost all the proofs of the LLT, the starting point in the proof of
Theorem 1.4 (and Theorem 2.1) is that for $k,N\in{\mathbb{N}}$ we have

(3.4)
$2\pi{\mathbb{P}}(S_{N}=k)=\int_{0}^{2\pi}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt.$

Denote ${\mathbb{T}}={\mathbb{R}}/2\pi{\mathbb{Z}}.$ Let

$\Phi_{N}(t)={\mathbb{E}}(e^{itS_{N}})=\prod_{n=1}^{N}\phi_{n}(t)\quad\text{where}\quad\phi_{n}(t)={\mathbb{E}}(e^{itX_{n}}).$

Divide ${\mathbb{T}}$ into intervals $I_{j}$ of small size $\delta$ such that
each interval contains at most one resonant point and this point is strictly
inside $I_{j}.$ We call an interval resonant if it contains a resonant point
inside. Then

(3.5)
$2\pi{\mathbb{P}}(S_{N}=k)=\sum_{j}\int_{I_{j}}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt.$

We will consider the integrals appearing in the above sum individually.

######  Lemma 3.3.

There are constants $C,c>0$ which depend only on ${\delta}$ and $K$ so that
for any non-resonant interval $I_{j}$ and $N\geq 1$ we have

$\int_{I_{j}}|\Phi_{N}(t)|dt\leq Ce^{-cV_{N}}.$

###### Proof.

Let ${\hat{q}}_{n},{\bar{q}}_{n}$ be the largest and the second largest values
of ${\mathbb{P}}(X_{n}=j)$ and let ${\hat{j}}_{n},{\bar{j}}_{n}$ be the
corresponding values. Note that

(3.6)
$\phi_{n}(t)={\hat{q}}_{n}e^{it{\hat{j}}_{n}}+{\bar{q}}_{n}e^{it{\bar{j}}_{n}}+\sum_{l\neq{\hat{j}}_{n},{\bar{j}}_{n}}{\mathbb{P}}(X_{n}=l)e^{itl}.$

Since $I_{j}$ is non resonant, the angle between $e^{it{\hat{j}}_{n}}$ and
$e^{it{\bar{j}}_{n}}$ is uniformly bounded from below. Indeed if this was not
the case we would have $t{\bar{j}}_{n}-t{\hat{j}}_{n}\approx 2\pi l_{n}$ for
some $l_{n}\in{\mathbb{Z}}.$ Then $t\approx\frac{2\pi l_{n}}{m_{n}}$ where
$m_{n}={\bar{j}}_{n}-{\hat{j}}_{n}$ contradicting the assumption that $I_{j}$
is non-resonant. Accordingly $\exists c_{1}>0$ such that
$\displaystyle\left|e^{it{\hat{j}}_{n}}+e^{it{\bar{j}}_{n}}\right|\leq
2-c_{1}.$ Therefore

$\left|{\hat{q}}_{n}e^{it{\hat{j}}_{n}}+{\bar{q}}_{n}e^{it{\bar{j}}_{n}}\right|\leq{({\hat{q}}_{n}-{\bar{q}}_{n})+{\bar{q}}_{n}\left|e^{it{\hat{j}}_{n}}+e^{it{\bar{j}}_{n}}\right|}\leq{\hat{q}}_{n}+{\bar{q}}_{n}-2c_{1}{\bar{q}}_{n}.$

Plugging this into (3.6), we conclude that $|\phi_{n}(t)|\leq
1-2c_{1}{\bar{q}}_{n}$ for $t\in I_{j}.$ Multiplying these estimates over $n$
and using that $1-x\leq e^{-x},\,x>0$, we get

$|\Phi_{N}(t)|\leq e^{-2c_{1}\sum_{n}{\bar{q}}_{n}}.$

Since $V(X_{n})\leq c_{2}{\bar{q}}_{n}$ for a suitable constant $c_{2}$ we can
rewrite the preceding as

(3.7) $|\Phi_{N}(t)|\leq e^{-c_{3}V_{N}},\,c_{3}>0.$

Integrating over $I_{j}$ we obtain the result. ∎

###  3.3. Prokhorov estimates

Next we consider the case where $I_{j}$ contains a nonzero resonant point
$t_{j}=\frac{2\pi l}{m}.$

######  Lemma 3.4.

There is a constant $c_{0}$ which depends only on $K$ so that for any nonzero
resonant point $t_{j}=2\pi l/m$ we have

(3.8) $\sup_{t\in I_{j}}|{\mathbb{E}}(e^{itS_{N}})|\leq e^{-c_{0}M_{N}(m)}.$

Thus, for any $r\geq 1$ there is a constant $R=R(r,K)$ such that if
$M_{N}(m)\geq R\ln V_{N}$, then the integral
$\int_{I_{j}}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt$ is $o(\sigma_{N}^{-r})$
uniformly in $k$, and so it only contributes to the error term.

###### Proof.

The estimate (3.8) follows from the arguments in [33], but for readers’
convenience we recall its proof. Let $X$ be an integer-valued random variable
so that $\|X\|_{L^{\infty}}\leq K$. Let $t_{0}=2\pi l/m$ be a nonzero resonant
point, where $\gcd(l,m)=1$. Let $t\in{\mathbb{T}}$ be so that

(3.9) $|t-t_{0}|\leq{\delta},$

where ${\delta}$ is a small positive number. Let $\phi_{X}(\cdot)$ denote the
characteristic function of $X$. Since $x\leq e^{x-1}$ for any real $x$ we have

$|\phi_{X}(t)|^{2}\leq e^{|\phi(t)|^{2}-1}.$

Next, we have

$|\phi_{X}(t)|^{2}-1=\phi(t)\phi(-t)-1=\sum_{j=-2K}^{2K}\sum_{s}\tilde{P}_{j}\left[\cos(t_{j})-1\right]$

where

$\tilde{P}_{j}=\sum_{s}{\mathbb{P}}(X=s){\mathbb{P}}(X=j+s).$

Fix some $-2K\leq j\leq 2K$. We claim that if ${\delta}$ in (3.9) is small
enough and $j\not\equiv 0\text{ mod }m$ then for each integer $w$ we have
$|t-2\pi w/j|\geq{\varepsilon}_{0}$ for some ${\varepsilon}_{0}>0$ which
depends only on $K$. This follows from the fact that $-2K\leq j\leq 2K$ and
that $2\pi w/j\not=t_{0}$ (and there is a finite number of resonant points).
Therefore,

$\cos(tj)-1\leq-{\delta}_{0}$

for some ${\delta}_{0}>0$. On the other hand, if $j=km$ for some integer $k$
then with $w=lk$ we have

$\displaystyle\cos(tj)-1=-2\sin^{2}(tj/2)=-2\sin^{2}\left((tj-2\pi
w)/2\right)$
$\displaystyle=-2\sin^{2}\left(j(t-t_{0})/2\right)\leq-{\delta}_{1}(t-t_{0})^{2}$

for some ${\delta}_{1}>0$ (assuming that $|t-t_{0}|$ is small enough). We
conclude that

$|\phi_{X}(t)|^{2}-1\leq-{\delta}_{0}\sum_{j\in
A}\tilde{P}_{j}-{\delta}_{1}(t-t_{0})^{2}\sum_{j\in B}\tilde{P}_{j}$

where $A=A(X)$ is the set of $j$’s between $-2K$ and $2K$ so that $j\not\equiv
0\text{ mod }m$ and $B=B(X)$ is its complement in ${\mathbb{Z}}\cap[-2K,2K]$.
Let $s_{0}$ be the most likely residue of $X$ mod $m$ and $s_{1}$ be the
second most likely residue class. Since

${\mathbb{P}}(X\equiv s_{0}\text{ mod
}m)\geq\frac{1}{m}\quad\text{and}\quad{\mathbb{P}}(X\equiv s_{1}\text{ mod
}m)=q_{m}(X)$

it follows that $\displaystyle\sum_{j\in
A}\tilde{P}_{j}\geq\frac{q_{m}(X)}{m}.$

Combining this with the trivial bound $\displaystyle\sum_{j\in
B}\tilde{P}_{j}\geq{\mathbb{P}}^{2}(X\equiv s_{0})\geq\frac{1}{m^{2}}$ we
obtain

$|\phi_{X}(t)|\leq\exp-\left[\frac{1}{2}\left(\frac{{\delta}_{0}q_{m}(X)}{m}+\frac{{\delta}_{1}(t-t_{0})^{2}}{m^{2}}\right)\right].$

Applying the above with $t_{0}=t_{j}$ and $X=X_{n}$, $1\leq n\leq N$ we get
that

(3.10) $|\Phi_{N}(t)|\leq e^{-c_{0}M_{N}(m)-\bar{c}_{0}N(t-t_{j})^{2}}\leq
e^{-c_{0}M_{N}(m)}$

where $c_{0}$ is some constant. ∎

######  Remark 3.5.

Using the first inequality in (3.10) and arguing as in [33, page 264], we can
deduce that there are positive constants $C,c_{1},c_{2}$ such that

(3.11) $\int_{I_{j}}|{\mathbb{E}}(e^{itS_{N}})|dt\leq
C\left(e^{-c_{1}{\sigma}_{N}}+\frac{e^{-c_{2}M_{N}(m)}}{{\sigma}_{N}}\right).$

This estimate plays an important role in the proof of the SLLT in [33], but
for our purposes a weaker bound (3.8) is enough. Note also that in order to
prove (3.8) we could have just used the trivial inequality $\cos(t_{j})-1\leq
0$ when $j\equiv 0\text{ mod }m$, but we have decided to present this part
from [33] in full.

######  Remark 3.6.

Let $d_{{\mathcal{R}}}$ be the minimal distance between two different resonant
points. Then, when ${\delta}<2d_{{\mathcal{R}}}$, we can take
${\delta}_{0}=1-\cos(d_{{\mathcal{R}}})$ in the proof of Lemma 3.4. Therefore,
we can take $c_{0}=\frac{1-\cos(d_{{\mathcal{R}}})}{4K}$ in (3.8). Hence Lemma
3.4 holds with $R(r,K)=\frac{r+1}{2c_{0}}.$

###  3.4.  Proof of Theorem 1.4

Fix some $r\geq 1$. Lemmas 3.3 and 3.4 show that if $M_{N}\geq R(r,K)\ln
V_{N}$, then all the integrals in the right hand side of (3.5) are of order
$o({\sigma}_{N}^{-r})$, except for the one corresponding to the resonant point
$t_{j}=0$. That is, for any ${\delta}>0$ small enough, uniformly in $k$ we
have

$2\pi{\mathbb{P}}(S_{N}=k)=\int_{-{\delta}}^{\delta}e^{-ihk}\Phi_{N}(h)dh+o({\sigma}_{N}^{-r}).$

In order to complete the proof of Theorem 1.4, we need to expand the above
integral. Making a change of variables $h\to h/{\sigma}_{N}$ and using
Proposition 3.1, we conclude that if ${\delta}$ is small enough then

$\int_{-{\delta}}^{\delta}e^{-ihk}\Phi_{N}(h)dh=$
${\sigma}_{N}^{-1}\int_{-{\delta}{\sigma}_{N}}^{{\delta}{\sigma}_{N}}e^{-ihk_{N}}e^{-h^{2}/2}\left(1+\sum_{u=1}^{r}\frac{A_{u,N}(h)}{{\sigma}_{N}^{k}}+\frac{h^{r+1}}{{\sigma}_{N}^{r+1}}O(1)\right)dh$

where $k_{N}=\left(k-{\mathbb{E}}(S_{N})\right)/{\sigma}_{N}$. Since the
coefficients of the polynomials $A_{u,N}$ are uniformly bounded in $N$, we can
just replace the above integral with the corresponding integral over all
${\mathbb{R}}$ (i.e. replace $\pm{\delta}{\sigma}_{N}$ with $\pm\infty$). Now
the Edgeworth expansions are achieved using that for any nonnegative integer
$q$ we have that $(it)^{q}e^{-t^{2}/2}$ is the Fourier transform of the $q$-th
derivative of $\textbf{n}(t)=\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}$ and that for
any real $a$,

(3.12)
$\int_{-\infty}^{\infty}e^{-iat}\widehat{\textbf{n}^{(q)}}(t)dt=\textbf{n}^{(q)}(a)=\frac{1}{\sqrt{2\pi}}(-1)^{q}H_{q}(a)e^{-a^{2}/2}$

where $H_{q}(a)$ is the $q$-th Hermite polynomial.

##  4\. Generalized Edgeworth expansions: Proof of Theorem Theorem 2.1

###  4.1. Contributions of resonant intervals.

Let $r\geq 1$. As in the proof of Theorem 1.4, our starting point is the
equality

(4.1)
$2\pi{\mathbb{P}}(S_{N}=k)=\int_{0}^{2\pi}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt=\sum_{j}\int_{I_{j}}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt$

which holds for any $k\in{\mathbb{N}}$. We will consider the integrals
appearing in the above sum individually. By Lemma 3.3 the integrals over non-
resonant intervals are of order $o({\sigma}_{N}^{-r})$, and so they can be
disregarded. Moreover, in §3.4 we have expanded the integral over the resonant
interval containing $0$. Now we will see that in the case $M_{N}<R(r,K)\ln
V_{N}$ the contribution of nonzero resonant points need not be negligible.

Let $t_{j}=\frac{2\pi l}{m}$ be a nonzero resonant point so that
$M_{N}(m)<R(r,K)\ln V_{N}$ and let $I_{j}$ be the resonant interval containing
it. Theorem 2.1 will follow from an appropriate expansion of the integral

$\int_{I_{j}}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt.$

We need the following simple result, which for readers’ convenience is
formulated as a lemma.

######  Lemma 4.1.

There exists ${\bar{\varepsilon}}>0$ so that for each $n\geq 1$ with
$q_{n}(m)\leq{\bar{\varepsilon}}$ we have $|\phi_{n}(t_{j})|\geq\frac{1}{2}$.
In fact, we can take ${\bar{\varepsilon}}=\frac{1}{4m}$.

###### Proof.

Recall that $t_{j}=2\pi l/m$. The lemma follows since for any random variable
$X$ we have $\displaystyle|{\mathbb{E}}(e^{it_{j}X})|=$

$\left|e^{it_{j}s(m,X)}-\sum_{u\not\equiv s(m,X)\text{ mod
m}}\big{(}e^{it_{j}s(m,X)}-e^{it_{j}u}\big{)}P(X\equiv u\text{ mod }m)\right|$
$\geq 1-2mq(m,X)$

where $s(m,X)$ is the most likely value of $X\text{ mod }m$ and $q(m,X)$ is
the second largest value among $P(X\equiv u\text{ mod }m)$, $u=0,1,2,...,m-1$.
Therefore, we can take ${\bar{\varepsilon}}=\frac{1}{4m}$. ∎

Next, set ${\bar{\varepsilon}}=\frac{1}{8K}$ and let
$N_{0}=N_{0}(N,t_{j},{\bar{\varepsilon}})$ be the number of all $n$’s between
$1$ to $N$ so that $q_{n}(m)>{\bar{\varepsilon}}$. Then $N_{0}\leq\frac{R\ln
V_{N}}{{\bar{\varepsilon}}}$ because $M_{N}(m)\leq R\ln V_{N}.$ By permuting
the indexes $n=1,2,...,N$ if necessary we can assume that $q_{n}(m)$ is non
increasing. Let $N_{0}$ be the largest number such that
$q_{N_{0}}\geq{\bar{\varepsilon}}.$ Decompose

(4.2) $\Phi_{N}(t)=\Phi_{N_{0}}(t)\Phi_{N_{0},N}(t)$

where $\displaystyle\Phi_{N_{0},N}(t)=\prod_{n=N_{0}+1}^{N}\phi_{n}(t).$

######  Lemma 4.2.

If the length ${\delta}$ of $I_{j}$ is small enough then for any $t=t_{j}+h\in
I_{j}$ and $N\geq 1$ we have

$\Phi_{N_{0},N}(t)=\Phi_{N_{0},N}(t_{j})\Phi_{N_{0},N}(h)\Psi_{N_{0},N}(h)$

where

$\Psi_{N_{0},N}(h)=\exp\left[O(M_{N}(m))\sum_{u=1}^{\infty}(O(1))^{u}h^{u}\right].$

###### Proof.

Denote

$\mu_{n}={\mathbb{E}}(X_{n}),\quad{\bar{X}}_{n}=X_{n}-\mu_{n},\quad{\bar{\phi}}_{n}(t)={\mathbb{E}}(e^{it{\bar{X}}_{n}}).$

Let $j_{n}(m)$ be the most likely residue mod $m$ for $X_{n}.$ Decompose

${\bar{X}}_{n}=s_{n}+Y_{n}+Z_{n}$

where $Z_{n}\in m{\mathbb{Z}}$, $s_{n}=j_{n}(m)-\mu_{n}$, so that
${\mathbb{P}}(Y_{n}\neq 0)\leq mq_{n}(m).$ Then for $t=t_{j}+h$,

(4.3)
${\bar{\phi}}_{n}(t)=e^{it_{j}s_{n}}{\mathbb{E}}\left(e^{it_{j}Y_{n}}e^{ih{\bar{X}}_{n}}\right)={\bar{\phi}}_{n}(t_{j})\psi_{n}(h)$

where

$\psi_{n}(h)=\left(1+\frac{ih{\mathbb{E}}(e^{it_{j}Y}{\bar{X}}_{n})-\frac{h^{2}}{2}{\mathbb{E}}(e^{it_{j}Y}({\bar{X}}_{n})^{2})+\dots}{{\mathbb{E}}(e^{it_{j}Y_{n}})}\right).$

Next, using that for any $x\in(-1,1)$ we have

$1+x=e^{\ln(1+x)}=e^{x-x^{2}/2+x^{3}/3-...}$

we obtain that for $h$ close enough to $0$,

(4.4)
$\psi_{n}(h)=\exp\left(\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\left(\frac{1}{{\mathbb{E}}(e^{it_{j}Y_{n}})}\sum_{q=1}^{\infty}\frac{(ih)^{q}}{q!}{\mathbb{E}}(e^{it_{j}Y_{n}}(\bar{X}_{n})^{q})\right)^{k}\right)$
$\displaystyle=\exp\left(\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\sum_{1\leq
j_{1},...,j_{k}}\frac{1}{({\mathbb{E}}(e^{it_{j}Y_{n}}))^{k}}\prod_{r=1}^{k}\frac{(ih)^{j_{r}}}{j_{r}!}{\mathbb{E}}(e^{it_{j}Y_{n}}(\bar{X}_{n})^{j_{r}})\right)$
$\displaystyle=\exp\left(\sum_{u=1}^{\infty}\left(\sum_{k=1}^{u}\frac{(-1)^{k+1}}{k}\sum_{j_{1}+...+j_{k}=u}\,\prod_{r=1}^{k}\frac{{\mathbb{E}}(e^{it_{j}Y_{n}}(\bar{X}_{n})^{j_{r}})}{{\mathbb{E}}(e^{it_{j}Y_{n}})j_{r}!}\right)(ih)^{u}\right).$

Observe next that

${\mathbb{E}}[e^{it_{j}Y_{n}}(\bar{X}_{n})^{j_{r}}]={\mathbb{E}}\left[(e^{it_{j}Y_{n}}-1)\big{(}(\bar{X}_{n})^{j_{r}}-{\mathbb{E}}[(\bar{X}_{n})^{j_{r}}]\big{)}\right]+{\mathbb{E}}[(\bar{X}_{n})^{j_{r}}]{\mathbb{E}}(e^{it_{j}Y_{n}})$

and so with $C=2K$, we have

$\frac{{\mathbb{E}}[e^{it_{j}Y_{n}}(\bar{X}_{n})^{j_{r}}]}{{\mathbb{E}}(e^{it_{j}Y_{n}})}=O(q_{n}(m))O(C^{j_{r}})+{\mathbb{E}}[(\bar{X}_{n})^{j_{r}}].$

Plugging this into (4.4) and using that for $h$ small enough,

$\exp\left[\sum_{u=1}^{\infty}\left(\sum_{k=1}^{u}\frac{(-1)^{k+1}}{k}\sum_{j_{1}+...+j_{k}=u}\prod_{r=1}^{k}\frac{{\mathbb{E}}({\bar{X}}_{n}^{j_{r}})}{j_{r}!}\right)(ih)^{u}\right]={\mathbb{E}}\left(e^{ih{\bar{X}}_{n}}\right)$

we conclude that

$\psi_{n}(h)={\mathbb{E}}(e^{ih\bar{X}_{n}})\exp\left[\sum_{u=1}^{\infty}(O(1))^{u}O(q_{n}(m))h^{u}\right].$

Therefore,

$\Phi_{N_{0},N}(t)=\Phi_{N_{0},N}(t_{j})\Phi_{N_{0},N}(h)\Psi_{N_{0},N}(h)$

where
$\displaystyle\Psi_{N_{0},N}(h)=\exp\left[O(M_{N}(m))\sum_{u=1}^{\infty}(O(1))^{u}h^{u}\right].$
∎

######  Remark 4.3.

We will see in §4.2 that the coefficients of the polynomials appearing in
Theorem 2.1 depend on the coefficients of the power series $\Psi_{N_{0},N}(h)$
(see, in particular, (4.22)). The first term in this series is $\displaystyle
ih\sum_{n={N_{0}+1}}^{N}a_{n,j}$, where

(4.5)
$a_{n,j}=\frac{{\mathbb{E}}[(e^{it_{j}Y_{n}}-1)\bar{X}_{n}]}{{\mathbb{E}}(e^{it_{j}Y_{n}})}=\frac{{\mathbb{E}}(e^{it_{j}X_{n}}\bar{X}_{n})}{{\mathbb{E}}(e^{it_{j}X_{n}})}$

while the second term is
$\displaystyle\frac{h^{2}}{2}\sum_{n={N_{0}+1}}^{N}b_{n,j}$, where

(4.6)
$b_{n,j}=\frac{{\mathbb{E}}[(e^{it_{j}Y_{n}}-1)\bar{X}_{n}]^{2}}{{\mathbb{E}}(e^{it_{j}Y_{n}})^{2}}-\frac{{\mathbb{E}}[(e^{it_{j}Y_{n}}-1)((\bar{X}_{n})^{2}-V(X_{n}))]}{{\mathbb{E}}(e^{it_{j}Y_{n}})}$
$=a_{n,j}^{2}-\frac{{\mathbb{E}}\big{(}e^{it_{j}X_{n}}(\bar{X}_{n})^{2}\big{)}}{{\mathbb{E}}(e^{it_{j}X_{n}})}.$

In Section 9 we will use (4.5) to compute the coefficients of the polynomials
from Theorem 2.1 in the case $r=2$, and (4.6) is one of the main ingredients
for the computation in the case $r=3$ (which will not be explicitly discussed
in this manuscript).

The next step in the proof of Theorem 2.1 is the following.

######  Lemma 4.4.

For $t=t_{j}+h\in I_{j}$ we can decompose

(4.7)
$\Phi_{N_{0}}(t)=\Phi_{N_{0}}(t_{j}+h)=\sum_{l=0}^{L}\frac{\Phi_{N_{0}}^{(l)}(t_{j})}{l!}h^{l}+O\left((h\ln
V_{N})^{L+1}\right).$

###### Proof.

The lemma follows from the observation that the derivatives of $\Phi_{N_{0}}$
satisfy $|\Phi_{N_{0}}^{(k)}(t)|\leq O(N_{0}^{k})\leq(C\ln V_{N})^{k}$. ∎

###  4.2. Completing the proof

Recall (4.1) and consider a resonant interval $I_{j}$ which does not contain
$0$ such that $M_{N}(m)\leq R\ln{\sigma}_{N}$. Set
$U_{j}=[-u_{j},v_{j}]=I_{j}-t_{j}$. Let $N_{0}$ be as described below Lemma
4.1. Denote

(4.8) $S_{N_{0},N}=S_{N}-S_{N_{0}},\,S_{0}=0,$
$V_{N_{0},N}=\text{Var}(S_{N}-S_{N_{0}})=V_{N}-V_{N_{0}}\quad\text{and}\quad{\sigma}_{N_{0},N}=\sqrt{V_{N_{0},N}}.$

Then

(4.9) $V_{N_{0},N}=V_{N}+O(\ln V_{N})=V_{N}(1+o(1)).$

Denote $h_{N_{0},N}=h/{\sigma}_{N_{0},N}.$ By (3.1), if $|h_{N,N_{0}}|$ is
small enough then

(4.10) ${\mathbb{E}}(e^{ih_{N_{0},N}S_{N_{0},N}})=$
$e^{ih_{N_{0},N}{\mathbb{E}}(S_{N_{0},N})}e^{-h^{2}/2}\left(1+\sum_{k=1}^{r}\frac{A_{k,N_{0},N}(h)}{{\sigma}_{N_{0},N}^{k}}+\frac{h^{r+1}}{{\sigma}_{N_{0},N}^{r+1}}O(1)\right)$

where $A_{k,N_{0},N}$ are polynomials with bounded coefficients (the degree of
$A_{k,N_{0},N}$ depends only on $k$). Let us now evaluate
$\displaystyle\int_{I_{j}}e^{-itk}{\mathbb{E}}(e^{itS_{N}})dt.$ By Lemma 4.2,

(4.11) $\int_{I_{j}}e^{-itk}\Phi_{N}(t)dt=$
$e^{-it_{j}k}\Phi_{N_{0},N}(t_{j})\int_{U_{j}}e^{-ihk}\Phi_{N_{0}}(t_{j}+h)\Phi_{N_{0},N}(h)\Psi_{N_{0},N}(h)\;dh.$

Therefore, it is enough to expand the integral on the RHS of (4.11). Fix a
large positive integer $L$ and plug (4.7) into (4.11). Note that for $N$ is
large enough, $h_{0}$ small enough and $|h|\leq h_{0}$, Proposition 3.2 and
(4.9) show that there exist positive constants $c_{0},c$ such that

(4.12) $|\Phi_{N_{0},N}(h)|=|{\mathbb{E}}(e^{ihS_{N_{0},N}})|\leq
e^{-c_{0}(V_{N}-V_{N_{0}})h^{2}}\leq e^{-cV_{N}h^{2}}.$

Thus, the contribution coming from the term $O\left((h\ln V_{N})^{L+1}\right)$
in the right hand side of (4.7) is at most of order

$V_{N}^{R{\delta}}(\ln
V_{n})^{L+1}\int_{-\infty}^{\infty}h^{L+1}e^{-cV_{N}h^{2}}dh$

where ${\delta}$ is the diameter of $I_{j}$. Changing variables
$x={\sigma}_{N}h$, where ${\sigma}_{N}=\sqrt{V_{N}}$ we get that the latter
term is of order $(\ln V_{n})^{L+1}{\sigma}_{N}^{-(L+1-2R{\delta})}$ and so
when $L$ is large enough we get that this term is $o({\sigma}_{N}^{-r-1})$
(alternatively, we can take $L=r$ and ${\delta}$ to be sufficiently small).
This means that it is enough to expand each integral of the form

(4.13) $\int_{U_{j}}e^{-ihk}h^{l}\Phi_{N_{0},N}(h)\Psi_{N_{0},N}(h)dh$

where $l=0,1,...,L$ (after changing variables the above integral is divided by
${\sigma}_{N_{0},N}^{l+1}$). Next, Lemma 4.2 shows that for any
$d\in{\mathbb{Z}}$ we have

(4.14)
$\Psi_{N_{0},N}(h)=1+\sum_{u=1}^{d}C_{w,N}h^{u}+h^{d+1}O(1+M_{N}(m)^{d+1})|V_{N}|^{O(|h|)},$

where $C_{w,N}=C_{w,N,t_{j}}$ are $O(M_{N}^{u}(m))=O((\ln V_{N})^{u})$. Note
that, with $a_{n,j}$ and $b_{n,j}$ defined in Remark 4.3, we have

(4.15) $C_{1,N}=i\sum_{n=N_{0}+1}^{N}a_{n,j}$

and

$C_{2,N}=\frac{1}{2}\sum_{n=N_{0}+1}^{N}b_{n,j}-\frac{1}{2}\left(\sum_{n=N_{0}+1}^{N}a_{n,j}\right)^{2}.$

Take $d$ large enough and plug (4.14) into (4.13). Using (4.12), we get again
that the contribution of the term

$h^{d+1}O(1+M_{N}(m)^{d+1})|V_{N}|^{O(|h|)}h^{l}\Phi_{N_{0},N}(h)$

to the above integral is $o({\sigma}_{N}^{-r})$. Thus, it is enough to expand
each term of the form

$\int_{U_{j}}e^{-ihk}h^{q}\Phi_{N_{0},N}(h)dh$

where $0\leq q\leq L+d$. Using (4.10) and making the change of variables $h\to
h/{\sigma}_{N_{0},N}$ it is enough to expand

(4.16)
$\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb{E}}[S_{N_{0},N}])/{\sigma}_{N_{0},N}}h^{q}e^{-h^{2}/2}\left(1+\sum_{w=1}^{r}\frac{A_{w,N_{0},N}(h)}{{\sigma}_{N_{0},N}^{w}}+\frac{h^{r+1}}{{\sigma}_{N_{0},N}^{r+1}}O(1)\right)\frac{dh}{{\sigma}_{N_{0},N}}.$

This is achieved by using that $(it)^{q}e^{-t^{2}/2}$ is the Fourier transform
of the $q$-th derivative of $\textbf{n}(t)=\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}$
and that for any real $a$,

(4.17)
$\int_{-\infty}^{\infty}e^{-iat}\widehat{\textbf{n}^{(q)}}(t)dt=\textbf{n}^{(q)}(a)=\frac{1}{\sqrt{2\pi}}(-1)^{q}H_{q}(a)e^{-a^{2}/2}$

where $H_{q}(a)$ is the $q$-th Hermite polynomial.

Note that in the above expansion we get polynomials in the variable
$\displaystyle
k_{N_{0},N}=\frac{k-{\mathbb{E}}[S_{N}-S_{N_{0}}]}{{\sigma}_{N,N_{0}}}$, not
in the variable $k_{N}=\frac{k-{\mathbb{E}}(S_{N})}{{\sigma}_{N}}$. Since
$k_{N_{0},N}=k_{N}{\alpha}_{N_{0},N}+O(\ln{\sigma}_{N}/{\sigma}_{N})$, where
${\alpha}_{N_{0},N}={\sigma}_{N}/{\sigma}_{N_{0},N}=O(1)$, the binomial
theorem shows that such polynomials can be rewritten as polynomials in the
variable $k_{N}$ whose coefficients are uniformly bounded in $N$. We also
remark that in the above expansions we get the exponential terms

$e^{-\frac{(k-a_{N_{0},N})^{2}}{2(V_{N}-V_{N_{0}})}}\quad\text{where}\quad
a_{N_{0},N}={\mathbb{E}}[S_{N}-S_{N_{0}}]$

and not $e^{-(k-a_{N})^{2}/2V_{N}}$ (as claimed in Theorem 2.1). In order to
address this fix some ${\varepsilon}<1/2$. Note that for $|k-a_{N_{0},N}|\geq
V_{N}^{\frac{1}{2}+{\varepsilon}}$ we have

$e^{-\frac{(k-a_{N_{0},N})^{2}}{2(V_{N}-V_{N_{0}})}}=o(e^{-cV_{N}^{2{\varepsilon}}})\quad\text{and}\quad
e^{-\frac{(k-a_{N_{0},N})^{2}}{2V_{N}}}=o(e^{-cV_{N}^{2{\varepsilon}}})\text{
for some }c>0.$

Since both terms are $o({\sigma}_{N}^{-s})$ for any $s$, it is enough to
explain how to replace $\displaystyle
e^{-\frac{(k-a_{N_{0},N})^{2}}{2(V_{N}-V_{N_{0}})}}$ with $\displaystyle
e^{-\frac{(k-a_{N})^{2}}{2V_{N}}}$ when $|k-a_{N_{0},N}|\leq
V_{N}^{\frac{1}{2}+{\varepsilon}}$ (in which case
$\displaystyle|k-a_{N}|=O(V_{N}^{\frac{1}{2}+{\varepsilon}})$). For such $k$’s
we can write

(4.18) $\exp\left[-\frac{(k-a_{N_{0},N})^{2}}{2(V_{N}-V_{N_{0}})}\right]=$
$\exp\left[-\frac{(k-a_{N_{0},N})^{2}}{2V_{N}}\right]\;\exp\left[-\frac{(k-a_{N_{0},N})^{2}V_{N_{0}}}{2V_{N}(V_{N}-V_{N_{0}})}\right].$

Since
$\displaystyle\frac{(k-a_{N_{0},N})^{2}V_{N_{0}}}{2V_{N}(V_{N}-V_{N_{0}})}=O\left(V_{N}^{-(1-3{\varepsilon})}\right)$,
for any $d_{1}$ we have

(4.19)
$\exp\left[-\frac{(k-a_{N_{0},N})^{2}V_{N_{0}}}{2V_{N}(V_{N}-V_{N_{0}})}\right]=$
$\sum_{j=0}^{d_{1}}\frac{V_{N_{0}}^{j}}{2^{j}(V_{N}-V_{N_{0}})^{j}j!}\left(\frac{(k-a_{N_{0},N})^{2}}{{\sigma}_{N}^{2}}\right)^{j}+O(V_{N}^{-(d_{1}+1)(2-3{\varepsilon})}).$

Note that (using the binomial formula) the first term on the above right hand
side is a polynomial of the variable $(k-a_{N})/{\sigma}_{N}$ whose
coefficients are uniformly bounded in $N$.

Next we analyze the first factor in the RHS of (4.18). As before, it is enough
to consider $k$’s such that $|k-a_{N}|\leq V_{N}^{\frac{1}{2}+{\varepsilon}}$
for a sufficiently small ${\varepsilon}.$ We have

(4.20) $\exp\left[-\frac{(k-a_{N,N_{0}})^{2}}{2V_{N}}\right]=$
$\exp\left[-\frac{(k-a_{N})^{2}}{2V_{N}}\right]\exp\left[-\frac{2(k-a_{N})a_{N_{0}}+a_{N_{0}}^{2}}{2V_{N}}\right].$

Note that
$\frac{(k-a_{N})a_{N_{0}}+a_{N_{0}}^{2}}{2V_{N}}=k_{N}\beta_{N_{0},N}+\theta_{N_{0},N}$,
where

$\beta_{N_{0},N}=\frac{a_{N_{0}}}{2{\sigma}_{N}}=O\left(\frac{\ln{\sigma}_{N}}{{\sigma}_{N}}\right)\;\text{
and
}\;\theta_{N_{0},N}=\frac{a_{N_{0}}^{2}}{2V_{N}}=O\left(\frac{\ln^{2}{\sigma}_{N}}{V_{N}}\right).$

Approximating $e^{\frac{(k-a_{N})a_{N_{0}}+a_{N_{0}}^{2}}{2V_{N}}}$ by a
polynomial of a sufficiently large degree $d_{2}$ in the variable
$\frac{(k-a_{N})a_{N_{0}}+a_{N_{0}}^{2}}{2V_{N}}$ completes the proof of
existence of polynomials $P_{a,b,N}$ claimed in the theorem (the Taylor
reminder in the last approximation is of order $\displaystyle
O\left(V_{N}^{-d_{2}(\frac{1}{2}-{\varepsilon})}\right)$, so we can take
$d_{2}=4(r+1)$ assuming that ${\varepsilon}$ is small enough).

Finally, let us show that the coefficients of the polynomials $P_{a,b,N}$
constructed above are uniformly bounded in $N$. In fact, we will show that for
each nonzero resonant point $t_{j}=2\pi l/m$, the coefficients of the
polynomials coming from integration over $I_{j}$ are of order

$O\left((1+M_{N}^{q_{0}}(m))e^{-M_{N}(m)}\right),$

where $q_{0}=q_{0}(r)$ depends only on $r$.

Observe that the additional contribution to the coefficients of the
polynomials coming from the transition between the variables $k_{N}$ and
$k_{N_{0},N}$ is uniformly bounded in $N$. Hence we only need to show that the
coefficients of the (original) polynomials in the variable $k_{N_{0},N}$ are
uniformly bounded in $N$. The possible largeness of these coefficient can only
come from the terms $C_{u,N,t_{j}}$, for $u=0,1,2,...,d$ which are of order
$M_{N}^{u}(m)$, respectively. However, the corresponding terms are multiplied
by terms of the form $\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})$ for
certain $\ell$’s which are uniformly bounded in $N$ (see also (4.22)). We
conclude that there are constants $W_{j}\in{\mathbb{N}}$ and
$a_{j}\in{\mathbb{N}}$ which depend only on $t_{j}$ and $r$ so that the
coefficients of the resulting polynomials are composed of a sum of at most
$W_{j}$ terms of order
$(M_{N}(m))^{a_{j}}\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})$, where
$\ell\leq E(r)$ for some $E(r)\in{\mathbb{N}}$. Next, we have

(4.21) $\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})=$
$\sum_{n_{1},\dots,n_{k}\leq
N_{0};\atop\ell_{1}+\dots+\ell_{k}=\ell}\gamma_{\ell_{1},\dots,\ell_{k}}\left(\prod_{q=1}^{k}\phi_{n_{q}}^{(\ell_{q})}(t_{j})\right)\left[\prod_{n\leq
N,\;n\neq n_{k}}\phi_{n}(t_{j})\right]$

where $\gamma_{\ell_{1},\dots,\ell_{k}}$ are bounded coefficients of
combinatorial nature. Using (3.8) we see that for each $n_{1},\dots,n_{k}$ the
product in the square brackets is at most $Ce^{-c_{0}M_{N}(m)+O(1)}$ for some
$C,c_{0}>0$. Hence

$|\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})|\leq\hat{C}N_{0}^{\ell}\;e^{-c_{0}M_{N}(m)},\,\,\hat{C}>0.$

Now, observe that the definition of $N_{0}$ gives
$M_{N}(m)\geq{\varepsilon}_{0}N_{0}$, ${\varepsilon}_{0}>0$. Therefore
$\displaystyle|\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})|\leq
C_{0}M_{N}^{\ell}(m)e^{-c_{0}M_{N}(m)}$, and so each one of the above
coefficients is of order $M_{N}^{\ell^{\prime}}(m)e^{-c_{0}M_{N}(m)}$ for some
$\ell^{\prime}$ which does not depend on $N$. ∎

######  Remark 4.5.

The transition between the variables $k_{N_{0},N}$ and $k_{N}$ changes the
monomials of the polynomials $P_{a,b,N}$, $a\not=0$ coming from integration
over $I_{j}$, for $t_{j}\not=0$ into monomials of the form
$\frac{c_{N}a_{N_{0}}^{j_{1}}{\sigma}_{N_{0}}^{j_{2}}k_{N}^{j_{3}}}{{\sigma}_{N}^{u}}$
for some bounded sequence $(c_{N})$, $j_{1},j_{2},j_{3}\geq 0$ and
$u\in{\mathbb{N}}$. As we have explained, the coefficients of these monomials
are uniformly bounded. Still, it seems more natural to consider such monomials
as part of the polynomial $P_{a,b+u,N}$. In this case we still get polynomials
with bounded coefficients since $a_{N_{0}}$ and ${\sigma}_{N_{0}}$ are both
$O(N_{0})$, $N_{0}=O(M_{N}(m))$ and $c_{N}$ contains a term of the form
$\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})$.

######  Remark 4.6.

As can be seen from the proof, the resulting expansions might contain terms
corresponding to ${\sigma}_{N}^{-s}$ for $s>r$. Such terms can be disregarded.
For $\frac{|k-a_{N}|}{{\sigma}_{N}}\leq V_{N}^{\varepsilon}$ this follows
because the coefficients of our exapansions are $O(1)$ and for
$\frac{|k-a_{N}|}{{\sigma}_{N}}\geq V_{N}^{\varepsilon}$ this follows from
(4.12). In practice, some of the polynomials $P_{a,b,N}$ with $b\leq r$ might
have coefficients which are $\displaystyle o({\sigma}_{N}^{b-r})$ (e.g. when
$b+u>r$ in the last remark) so they also can be disregarded. The question when
the terms $P_{a,b,N}$ may be disregarded is in the heart of the proof of
Theorem 1.5 given in the next section.

###  4.3. A summary

The proofs of Proposition 1.2, Theorem 1.3 and Theorem 1.5 will be based on
careful analysis of the formulas of the polynomials from Theorem 2.1. For this
purpose, it will be helpful to summarize the main conclusions from the proof
of Theorem 2.1. Let $r\geq 1$ and $t_{j}=2\pi l/m$ be a nonzero resonant
point. Then the arguments in the proof of Theorem 2.1 yield that the
contribution to the expansion coming from $t_{j}$ is

(4.22) $\textbf{C}_{j}(k):=$ $e^{-it_{j}k}\Phi_{N_{0},N}(t_{j})\sum_{s\leq
r-1}\left(\sum_{u+l=s}\frac{\Phi_{N_{0}}^{(l)}(t_{j})C_{u,N}}{l!}\right)\int_{U_{j}}e^{-ihk}h^{s}\Phi_{N_{0},N}(h)dh$

where $U_{j}=I_{j}-t_{j}$, $C_{u,N}$ are given by (4.14) and $C_{0,N}=1$. When
$t_{j}=0$ then it is sufficient to consider only $s=0$, $N_{0}=0$ and the
contribution is just the integral

$\int_{-{\delta}}^{\delta}e^{-ihk}\Phi_{N}(h)dh$

where ${\delta}$ is small enough. As in (4.16), changing variables we can
replace the integral corresponding to $h^{s}$ with

(4.23)
$\displaystyle{\sigma}_{N_{0},N}^{-s-1}\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb{E}}[S_{N_{0},N}])/{\sigma}_{N_{0},N}}h^{s}e^{-h^{2}/2}$
$\displaystyle\times\left(1+\sum_{w=1}^{r}\frac{A_{w,N_{0},N}(h)}{{\sigma}_{N_{0},N}^{w}}+\frac{h^{r+1}}{{\sigma}_{N_{0},N}^{r+1}}O(1)\right)dh.$

After that was established, the proof was completed using (4.17) and some
estimates whose whose purpose was to make the transition between the variables
$k_{N_{0},N}$ and $k_{N}$.

##  5\. Uniqueness of trigonometric expansions.

In several proofs we will need the following result.

######  Lemma 5.1.

Let $r\geq 1$ and $d\geq 0$. Set ${\mathcal{R}}_{0}={\mathcal{R}}\cup\\{0\\}$
where ${\mathcal{R}}$ is the set of nonzero resonant points. For any
$t_{j}\in{\mathcal{R}}_{0}$, let $A_{0,N}(t_{j})$,…,$A_{d,N}(t_{j})$ be
sequences so that, uniformly in $k$ such that $\displaystyle
k_{N}=\frac{k-{\mathbb{E}}(S_{N})}{{\sigma}_{N}}=O(1)$ we have

$\sum_{t_{j}\in{\mathcal{R}}_{0}}e^{-it_{j}k}\left(\sum_{m=0}^{d}k_{N}^{m}A_{m,N}(t_{j})\right)=o({\sigma}_{N}^{-r}).$

Then for all $m$ and $t_{j}$

(5.1) $A_{m,N}(t_{j})=o({\sigma}_{N}^{-r}).$

In particular the polynomials from the definition of the (generalized)
Edgeworth expansions are unique up to terms of order $o({\sigma}_{N}^{-r})$.

###### Proof.

The proof is by induction on $d$. Let us first set $d=0$. Then, for any
$k\in{\mathbb{N}}$ we have

(5.2)
$\sum_{t_{j}\in{\mathcal{R}}_{0}}e^{-it_{j}k}A_{0,N}(t_{j})=o({\sigma}_{N}^{-r}).$

Let $T$ be the number of nonzero resonant points, and let us relabel them as
$\\{x_{1},...,x_{T}\\}$. Consider the vector

$\mathfrak{A}_{N}=(A_{0,N}(0),A_{0,N}(x_{1}),...,A_{0,N}(x_{T})).$

Let ${\mathcal{V}}$ be the transpose of the Vandermonde matrix of the distinct
numbers ${\alpha}_{j}=e^{-ix_{j}},j=0,1,2,...,T$ where $x_{0}:=0$. Then
${\mathcal{V}}$ is invertible and by considering $k=0,1,2,...,T$ in (5.2) we
see that (5.2) holds true if and only if

$\mathfrak{A}_{N}={\mathcal{V}}^{-1}o({\sigma}_{N}^{-r})=o({\sigma}_{N}^{-r}).$

Alternatively, let $Q$ be the least common multiple of the denominators of
$t_{j}\in{\mathcal{R}}$. Let $a_{N}(p)=A_{0,N}(2\pi p/Q)$ if $2\pi p/Q$ is a
resonant point and $0$ otherwise. Then for $m=0,1,...,Q-1$ we have

$\hat{a}_{N}(m):=\sum_{p=0}^{Q-1}a_{N}(p)e^{-2\pi pm/Q}=o({\sigma}_{N}^{-r}).$

Therefore, by the inversion formula of the discrete Fourier transform,

$a_{N}(p)=Q^{-1}\sum_{m=0}^{Q-1}\hat{a}_{N}(m)e^{2\pi
imp/Q}=o({\sigma}_{N}^{-r}).$

Assume now that the theorem is true for some $d\geq 0$ and any sequences
functions $A_{0,N}(t_{j}),...,A_{d,N}(t_{j})$. Let
$A_{0,N}(t_{j}),...,A_{d+1,N}(t_{j})$ be sequences so that uniformly in $k$
such that $\displaystyle
k_{N}:=\frac{k-{\mathbb{E}}(S_{N})}{{\sigma}_{N}}=O(1)$ we have

(5.3)
$\sum_{t_{j}\in{\mathcal{R}}_{0}}e^{-it_{j}k}\left(\sum_{m=0}^{d+1}k_{N}^{m}A_{m,N}(t_{j})\right)=o({\sigma}_{N}^{-r}).$

Let us replace $k$ with $k^{\prime}=k+[{\sigma}_{N}]Q$, where $Q$ is the least
common multiply of all the denominators of the nonzero $t_{j}$’s. Then
$e^{-it_{j}k}=e^{-it_{j}k^{\prime}}$. Thus,

$\sum_{t_{j}\in{\mathcal{R}}_{0}}e^{-it_{j}k}\left(\sum_{m=0}^{d+1}(k_{N}^{\prime
m}-k_{N}^{m})A_{m,N}(t_{j})\right)=o({\sigma}_{N}^{-r}).$

Set $L_{N}=[{\sigma}_{N}]Q/{\sigma}_{N}\thickapprox Q$. Then the LHS above
equals

$L_{N}\sum_{t_{j}\in{\mathcal{R}}_{0}}e^{-it_{j}k}\left(\sum_{s=0}^{d}k_{N}^{s}{\mathcal{A}}_{s,N}(t_{j})\right)$

where

${\mathcal{A}}_{s,N}(t_{j})=\sum_{m=s+1}^{d+1}A_{m,N}(t_{j})L_{N}^{m-s-1}.$

By the induction hypothesis we get that

${\mathcal{A}}_{s,N}(t_{j})=o({\sigma}_{N}^{-r})$

for any $s=0,1,...,d$. In particular

${\mathcal{A}}_{d,N}(t_{j})=A_{d+1,N}(t_{j})=o({\sigma}_{N}^{-r}).$

Substituting this into (5.3) we can disregard the last term
$A_{d+1,N}(t_{j})$. Using the induction hypothesis with
$A_{0,N}(t_{j}),A_{1,N}(t_{j}),...,A_{d,N}(t_{j})$ we obtain (5.1). ∎

##  6\. First order expansions

In this section we will consider the case $r=1$. By (4.22) and (4.23), we see
that the contribution coming from the integral over $I_{j}$ is

${\sigma}_{N_{0},N}^{-1}e^{-it_{j}k}\Phi_{N}(t_{j})\sqrt{2\pi}e^{-k_{N_{0},N}^{2}/2}+o({\sigma}_{N}^{-1})$

where $k_{N_{0},N}=(k-{\mathbb{E}}(S_{N_{0},N}))/{\sigma}_{N_{0},N}$. Now,
using the arguments at the end of the proof of Theorem 2.1 when $r=1$ we can
just replace $e^{-k_{N_{0},N}^{2}/2}$ with
$e^{-(k-{\mathbb{E}}(S_{N}))^{2}/2V_{N}}$ (since it is enough to consider the
case when $k_{N_{0},N}$ and $k_{0,N}$ are of order $V_{N}^{\varepsilon}$).
Therefore, taking into account that
${\sigma}_{N_{0},N}^{-1}-{\sigma}_{N}^{-1}=O({\sigma}_{N}^{-2}N_{0})$ we get
that

(6.1) $\sqrt{2}\pi{\mathbb{P}}(S_{N}=k)=$
$\left(1+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})\right){\sigma}_{N}^{-1}e^{-(k-{\mathbb{E}}[S_{N}])^{2}/2V_{N}}+o(\sigma_{N}^{-1}).$

Here ${\mathcal{R}}$ is the set of all nonzero resonant points $t_{j}=2\pi
l_{j}/m_{j}$. Indeed for the contribution of the resonant points satisfying
$M_{N}(m_{j})\leq R(r,K)\ln V_{N}$ is analyzed in §4.2. The contribution of
the other nonzero resonant points $t$ is $o({\sigma}_{N}^{-1})$ due to (3.8)
in Section 3. In particular, (3.8) implies that
$\Phi_{N}(t)=o({\sigma}_{N}^{-1})$ so adding the points with $M_{N}(m_{j})\geq
R(r,K)\ln V_{N}$ only changes the sum in the RHS of (6.1) by
$o(\sigma_{N}^{-1}).$

######  Corollary 6.1.

The local limit theorem holds if and only if $\displaystyle\max_{t\in
R}|\Phi_{N}(t)|=o(1)$.

###### Proof.

It follows from (6.1) that the LLT holds true if and only if for any $k$ we
have

$\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})=o(1).$

Now, the corollary follows from Lemma 5.1. ∎

Before proving Theorem 1.3 we recall a standard fact which will also be useful
in the proofs of Theorems 1.6 and 1.8.

######  Lemma 6.2.

Let $\\{\mu_{N}\\}$ be a sequence of measures probability measures on
${\mathbb{Z}}/m{\mathbb{Z}}$ and $\\{\gamma_{N}\\}$ be a positive sequence.
Then $\mu_{N}(a)=\frac{1}{m}+O(\gamma_{N})$ for all
$a\in{\mathbb{Z}}/m{\mathbb{Z}}$ if and only iff
$\hat{\mu}_{N}(b)=O(\gamma_{N})$ for all
$b\in\left({\mathbb{Z}}/m{\mathbb{Z}}\right)\setminus\\{0\\}$ where
$\hat{\mu}$ is the Fourier transform of $\mu.$

###### Proof.

If $\mu_{N}(a)=\frac{1}{m}+O(\gamma_{n})$ then

$\hat{\mu}_{N}(b)=\sum_{a=0}^{m-1}\mu_{N}(a)e^{2\pi
iab/m}=\sum_{a=0}^{m-1}\frac{1}{m}e^{2\pi iab/m}+O(\gamma_{N})=O(\gamma_{N}).$

Next $\hat{\mu}_{N}(0)=1$ since $\mu_{N}$ are probabilities. Hence if
$\hat{\mu}_{N}(b)=O(\gamma_{N})$ for all
$b\in\left({\mathbb{Z}}/m{\mathbb{Z}}\right)\setminus\\{0\\}$ then

$\mu_{N}(a)=\frac{1}{m}\sum_{b=0}^{m-1}\hat{\mu}_{N}(b)e^{-2\pi
iba/m}=\frac{1}{m}\left[1+\sum_{b=1}^{m-1}\hat{\mu}_{N}(b)e^{-2\pi
iba/m}\right]=\frac{1}{m}+O(\gamma_{N})$

as claimed. ∎

###### Proof of Theorem 1.3.

The equivalence of conditions (b) and (c) comes from the fact that for non-
resonant points the characteristic function decays faster than any power of
$\sigma_{N}$ (see (3.7)).

The equivalence of (a) and (c) is due to Corollary 6.1. Finally, the
equivalence between (c) and (d) comes from Lemma 6.2. ∎

######  Remark 6.3.

Theorem 1.3 can also be deduced from [7, Corollary 1.4]. Indeed the corollary
says that either the LLT holds or there is an integer $h\in(0,2K)$ and a
bounded sequence $\\{a_{N}\\}$ such that the limit

${\mathbf{p}}(j)=\lim_{N\to\infty}{\mathbb{P}}(S_{N}-a_{N}=j\text{ mod }h)$

exists and moreover if $k-a_{n}\equiv j\text{ mod }h$ then

$\sigma_{N}{\mathbb{P}}(S_{N}=k)={\mathbf{p}}(j)h\mathfrak{g}\left(\frac{k-{\mathbb{E}}(S_{N})}{\sigma_{N}}\right)+o\left(\sigma_{N}^{-1}\right).$

Thus in the second case the LLT holds iff ${\mathbf{p}}(j)=\frac{1}{h}$ for
all $j$ which is equivalent to $S_{N}$ being asymptotically uniformly
distributed mod $h$ and also to the Fourier transform of ${\mathbf{p}}(j)$
regarded as the measure on ${\mathbb{Z}}/(h{\mathbb{Z}})$ being the $\delta$
measure at 0. Thus the conditions (a), (c) and (d) of the theorem are
equivalent. Also by the results of [7, Section 2] (see also [§3.3.2][10]) if
${\mathbb{E}}\left(e^{i\xi S_{N}}\right)$ does not converge to 0 for some non
zero $\xi$ then $\displaystyle\left(\frac{2\pi}{\xi}\right){\mathbb{Z}}\bigcap
2\pi{\mathbb{Z}}$ is a lattice in ${\mathbb{R}}$ which implies that $\xi$ is
resonant, so condition (b) of the theorem is also equivalent to the other
conditions.

###### Proof of Proposition 1.2..

Let $S_{N}$ satisfy LLT. Fix $m\in\mathbb{N}$ and suppose that
$\displaystyle\sum_{n}q_{n}(m)<\infty.$ Let $s_{n}$ be the most likely residue
of $X_{n}$ mod $m$. Then for $t=\frac{2\pi l}{m}$ we have

$\phi_{n}(t)=e^{its_{n}}-\sum_{j\not\equiv
s_{n}\;\text{mod}\;m}{\mathbb{P}}(X_{n}\equiv j\;\text{mod
m})\left(e^{its_{n}}-e^{itj}\right),$

so that $1\geq|\phi_{n}(t)|\geq 1-2mq_{n}(m).$ It follows that for each
${\varepsilon}>0$ there is $N({\varepsilon})$ such that
$\displaystyle\left|\prod_{n=N({\varepsilon})+1}^{\infty}\phi_{n}(t)\right|>1-{\varepsilon}.$
Applying this for ${\varepsilon}=\frac{1}{2}$ we have

(6.2) $\frac{1}{2}\leq\liminf_{N\to\infty}\left|\Phi_{N(1/2),N}(t)\right|\neq
0.$

On the other hand the LLT implies that

(6.3) $\lim_{N\to\infty}\Phi_{N}(t)=0.$

Since $\Phi_{N}=\Phi_{N(1/2)}\Phi_{N(1/2),N}$, (6.2) and (6.3) imply that
$\Phi_{N(1/2)}(t)=0.$ Since $\displaystyle\Phi_{N(1/2)}\left(\frac{2\pi
l}{m}\right)=\prod_{n=1}^{N(1/2)}\phi_{n}\left(\frac{2\pi l}{m}\right)$ we
conclude that there exists $n_{l}\leq N(1/2)$ such that
$\phi_{n_{l}}(\frac{2\pi l}{m})=0.$ Hence $Y=X_{n_{1}}+X_{n_{2}}+\dots
X_{n_{m-1}}$ satisfies ${\mathbb{E}}\left(e^{2\pi i(k/m)Y}\right)=0$ for
$k=1,\dots m-1.$ By Lemma 6.2 both $Y$ and $S_{N}$ for $N\geq N(1/2)$ are
uniformly distributed. This proves the proposition. ∎

##  7\. Characterizations of Edgeworth expansions of all orders.

###  7.1. Derivatives of the non-perturbative factor.

Next we prove the following result.

######  Proposition 7.1.

Fix $r\geq 1,$ and assume that $M_{N}\leq R(r,K)\ln{\sigma}_{N}$ (possibly
along a subsequence). Then Edgeworth expansions of order $r$ hold true (i.e.
(1.2) holds for such $N$’s) iff for each $t_{j}\in{\mathcal{R}}$ and
$0\leq\ell<r$ (along the underlying subsequence) we have

(7.1)
${\sigma}_{N}^{r-\ell-1}\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})=o(1).$

###### Proof.

First, in view of (4.22) and (4.12), it is clear that the condition (7.1) is
sufficient for expansions of order $r$.

Let us now prove that the condition (7.1) is necessary for the expansion of
order $r.$

We will use induction on $r$. For $r=1$ (see (6.1)) our expansions read

${\mathbb{P}}(S_{N}=k)={\sigma}_{N}^{-1}e^{-k_{N}^{2}/2}\left[1+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})\right]+o({\sigma}_{N}^{-1}).$

Therefore if

${\mathbb{P}}(S_{N}=k)={\sigma}_{N}^{-1}e^{-k_{N}^{2}/2}P_{N}(k_{N})+o({\sigma}_{N}^{-1})$

for some polynomial $P_{N}$ then Lemma 5.1 tells us that, in particular
$\Phi_{N}(t_{j})=o(1)$ for each $t_{j}\in{\mathcal{R}}$.

Let us assume now that the necessity part in Proposition 7.1 holds for
$r^{\prime}=r-1$ and prove that it holds for $r$. We will use the following
lemma.

######  Lemma 7.2.

Assume that for some $t_{j}\in{\mathcal{R}}$,

(7.2)
${\sigma}_{N}^{r-2-l}\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(l)}(t_{j})=o(1),\,l=0,1,...,r-2.$

Then, up to an $o({\sigma}_{N}^{-r})$ error term, the contribution of $t_{j}$
to the generalized Edgeworh expansions of order $r$ is

(7.3)
$e^{-it_{j}k}e^{-k_{N}^{2}/2}\left(\frac{\Phi_{N_{0}}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{\mathscr{H}_{N,q}(k_{N})}{{\sigma}_{N}^{q}}\right)$

with

(7.4) $\mathscr{H}_{N,q}(x)=\mathscr{H}_{N,q}(x;t_{j})=$
${\mathcal{H}}_{N,q,1}(x)+{\mathcal{H}}_{N,q,2}(x)+{\mathcal{H}}_{N,q,3}(x)+{\mathcal{H}}_{N,q,4}(x)$

where

${\mathcal{H}}_{N,q,1}(x)=\frac{(i)^{q-1}H_{q-1}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-1)}(t_{j})}{(q-1)!},$
${\mathcal{H}}_{N,q,2}(x)=\frac{(i)^{q-1}H_{q-1}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-2)}(t_{j})C_{1,N,t_{j}}}{(q-2)!},$
${\mathcal{H}}_{N,q,3}(x)=\frac{a_{N_{0}}(i)^{q-2}H^{\prime}_{q-2}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-2)}(t_{j})}{(q-2)!},$
${\mathcal{H}}_{N,q,4}(x)=-\frac{xa_{N_{0}}(i)^{q-2}H_{q-2}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-2)}(t_{j})}{(q-2)!},$

and $H_{q}$ are Hermite polynomials.

Here $C_{1,N,t_{j}}$ is given by (4.15) when $M_{N}(m)\leq
R(K,r)\ln{\sigma}_{N}$, and $C_{1,N,t_{j}}=0$ when
$M_{N}(m)>R(K,r)\ln{\sigma}_{N}.$ (Note that in either case
$C_{1,N,t_{j}}=O(M_{N}(m))=O(\ln{\sigma}_{N})$).

As a consequence, when the Edgeworth expansions of order $r$ hold true and
(7.2) holds, then uniformly in $k$ so that $k_{N}=O(1)$ we have

(7.5)
$\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{\mathscr{H}_{N,q}(k_{N};t_{j})}{{\sigma}_{N}^{q}}=o({\sigma}_{N}^{-r}).$

The proof of the lemma will be given in §7.2 after we finish the proof of
Proposition 7.1.

By the induction hypothesis the condition (7.2) holds true. Let us prove now
that for $\ell=0,1,2,...,r-1$ and $t_{j}\in{\mathcal{R}}$ we have

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})=o({\sigma}_{N}^{-r+1+\ell}).$

Let us write

$\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{\mathscr{H}_{N,q}(k_{N})}{{\sigma}_{N}^{q}}=\sum_{m=0}^{r-1}k_{N}^{m}A_{m,N}(t_{j}).$

Applying Lemmas 5.1 and 7.2 we get that

$A_{m,N}(t_{j})=o({\sigma}_{N}^{-r})$

for each $0\leq m\leq r-1$ and $t_{j}\in{\mathcal{R}}$.

Fix $t_{j}\in{\mathcal{R}}$. Using Lemma 7.2 and the fact that the Hermite
polynomials $H_{u}$ have the same parity as $u$ and that their leading
coefficient is $1$ we have

(7.6) $\displaystyle
A_{r-1,N}(t_{j})={\sigma}_{N}^{-r}(i)^{r-2}\Bigg{(}i\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-1)}(t_{j})/(r-1)$
$\displaystyle+\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-2)}(t_{j})(iC_{1,N}-a_{N_{0}})\Bigg{)}=o({\sigma}_{N}^{-r})$

and

(7.7) $\displaystyle\quad
A_{r-2,N}(t_{j})={\sigma}_{N}^{-r+1}(i)^{r-3}\Bigg{(}i\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-2)}(t_{j})/(r-2)$
$\displaystyle+\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-3)}(t_{j})(iC_{1,N}-a_{N_{0}})\Bigg{)}=o({\sigma}_{N}^{-r}).$

Since $\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-3)}(t_{j})=o({\sigma}_{N}^{-1})$,
(7.7) yields

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-2)}(t_{j})=o({\sigma}_{N}^{-1}\ln{\sigma}_{N}).$

Plugging this into (7.6) we get

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-1)}(t_{j})=o(1).$

Therefore we can just disregard $\mathscr{H}_{r,N}(k_{N};t_{j})$ since its
coefficients are of order $o({\sigma}_{N}^{-r})$. Since the term
${\mathcal{H}}_{r,N}(k_{N};t_{j})$ no longer appears, repeating the above
arguments with $r-1$ in place of $r$ we have

$\displaystyle
A_{r-3,N}(t_{j})={\sigma}_{N}^{-r+2}(i)^{r-4}\Bigg{(}i\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-3)}(t_{j})/(r-3)$
$\displaystyle+\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-4)}(t_{j})(iC_{1,N}-a_{N_{0}})\Bigg{)}=o({\sigma}_{N}^{-r}).$

Since $\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-4)}(t_{j})=o({\sigma}_{N}^{-2})$,
the above asymptotic equality yields that

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-3)}(t_{j})=o({\sigma}_{N}^{-2}\ln{\sigma}_{N}).$

Plugging this into (7.7) we get

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(r-2)}(t_{j})=o({\sigma}_{N}^{-1}).$

Hence, we can disregard also the term $\mathscr{H}_{r-1,N}(k_{N};t_{j})$.
Proceeding this way we get that
$\displaystyle\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})=o({\sigma}_{N}^{\ell+1-r})$
for any $0\leq\ell<r$. ∎

Before proving Lemma 7.2, let us state the following result, which is a
consequence of Proposition 7.1 and (3.8).

######  Corollary 7.3.

Suppose that for each nonzero resonant point $t$ we have
$\displaystyle\inf_{n}|\phi_{n}(t)|>0$. Then for any $r$, the sequence $S_{N}$
obeys Edgeworth expansions of order $r$ if and only if
$\Phi_{N}(t)=o({\sigma}_{N}^{1-r})$ for each nonzero resonant point $t$.

###  7.2. Proof of Lemma 7.2.

###### Proof.

First, because of (7.2), for each $0\leq s\leq r-1,$ the terms indexed by
$l<s-1$ in (4.22), are of order $o({\sigma}_{N}^{-r})$ and so they can be
disregarded. Therefore, we need only to consider the terms indexed by $l=s$
and $l=s-1$. For such $l$, using again (7.2) we can disregard all the terms in
(4.23) indexed by $w\geq 1$, since the resulting terms are of order
$o({\sigma}_{N}^{-w-r}\ln{\sigma}_{N})=o({\sigma}_{N}^{-r})$. Now, since
${\sigma}_{N_{0},N}^{-1}-{\sigma}_{N}^{-1}=O(V_{N_{0}}/{\sigma}_{N}^{3})$ we
can replace ${\sigma}_{N_{0},N}$ with ${\sigma}_{N}^{-1}$ in (4.22), as the
remaining terms are of order $o({\sigma}_{N}^{-r-1})$. Therefore, using (4.17)
we get the following contributions from $t_{j}\in{\mathcal{R}}$,

$e^{-it_{j}k}\;e^{-k_{N_{0},N}^{2}/2}\left(\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{{\mathcal{H}}_{N,q}(k_{N_{0},N})}{{\sigma}_{N}^{q}}\right)$

where
${\mathcal{H}}_{N,q}(x)={\mathcal{H}}_{N,q,1}(x)+{\mathcal{H}}_{N,q,2}(x)$ and
${\mathcal{H}}_{N,q,j},j=1,2$ are defined after (7.4). Note that when $x=O(1)$
and $q<r$,

(7.8)
$\frac{{\mathcal{H}}_{N,q,1}(x)}{{\sigma}_{N}^{q}}=o({\sigma}_{N}^{-r+1})\,\,\text{
and
}\,\,\frac{{\mathcal{H}}_{N,q,2}(x)}{{\sigma}_{N}^{q}}=o({\sigma}_{N}^{-r}\ln{\sigma}_{N}).$

while when $q=r$,

(7.9)
$\frac{{\mathcal{H}}_{N,r,1}(x)}{{\sigma}_{N}^{r}}=O({\sigma}_{N}^{-r}\ln{\sigma}_{N})\,\,\text{
and
}\,\,\frac{{\mathcal{H}}_{N,r,2}(x)}{{\sigma}_{N}^{r}}=o({\sigma}_{N}^{-r}\ln{\sigma}_{N}).$

Next

$k_{N_{0},N}=(1+\rho_{N_{0},N})k_{N}+\frac{a_{N_{0}}}{{\sigma}_{N}}+\theta_{N_{0},N}$

where
$\rho_{N_{0},N}={\sigma}_{N}/{\sigma}_{N_{0},N}-1=O(\ln{\sigma}_{N}/{\sigma}^{2}_{N})$
and

$\theta_{N_{0},N}=a_{N_{0}}\left(\frac{1}{{\sigma}_{N_{0},N}}-\frac{1}{{\sigma}_{N}}\right)=O(\ln^{2}{\sigma}_{N}/{\sigma}_{N}^{3}).$

Hence, when $|k_{N_{0},N}|\leq{\sigma}_{N}^{{\varepsilon}}$ (and so
$\displaystyle k_{N}=O({\sigma}_{N}^{{\varepsilon}})$) for some
${\varepsilon}>0$ small enough then for each $m\geq 1$ we have

$k_{N_{0},N}^{m}=k_{N}^{m}+mk_{N}^{m-1}a_{N_{0}}/{\sigma}_{N}+o({\sigma}_{N}^{-1}).$

Therefore, (7.8) and (7.9) show that upon replacing $H_{q-1}(k_{N_{0},N})$
with $H_{q-1}(k_{N})$ the only additional term is

$\frac{a_{N_{0}}(i)^{q-1}H^{\prime}_{q-1}(k_{N})\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-1)}(t_{j})}{(q-1)!{\sigma}_{N}^{q+1}}$

for $q=2,3,...,r-1$. We thus get that the contribution of $t_{j}$ is

$e^{-it_{j}k}\;e^{-k_{N_{0},N}^{2}/2}\left(\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{{\mathcal{C}}_{N,q}(k_{N})}{{\sigma}_{N}^{q}}\right)$

where

${\mathcal{C}}_{N,q}(x)={\mathcal{H}}_{N,q}(x)+\frac{a_{N_{0}}(i)^{q-2}H^{\prime}_{q-2}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-2)}(t_{j})}{(q-2)!}.$

Note that ${\mathcal{C}}_{N,2}(\cdot)={\mathcal{H}}_{N,2}(\cdot)$. Finally, we
can replace $e^{-k_{N_{0},N}^{2}/2}$ with

$(1-k_{N}a_{N_{0}}/{\sigma}_{N})e^{-k_{N}^{2}/2}$

since all other terms in the transition between $e^{-k_{N_{0},N}^{2}/2}$ to
$e^{-k_{N}^{2}/2}$ are of order $o({\sigma}_{N}^{-1})$ (see (4.18) and
(4.19)). The term $-k_{N}a_{N_{0}}/{\sigma}_{N}$ shifts the $u$-th order term
to the $u+1$-th term, $u=1,2,...,r-1$ multiplied by $-k_{N}a_{N_{0}}$. Next,
relying on (7.8) and (7.9) we see that after multiplied by
$k_{N}a_{N_{0}}/{\sigma}_{N}$, the second term ${\mathcal{H}}_{N,q,2}(k_{N})$
from the definition of ${\mathcal{H}}_{N,q}(k_{N})$ is of order
$o({\sigma}_{N}^{-r-1}\ln^{2}{\sigma}_{N}){\sigma}_{N}^{q}$ and so this
product can be disregarded. Similarly, we can ignore the additional
contribution coming from multiplying the second term from the definition of
${\mathcal{C}}_{N,q}(k_{N})$ by $-k_{N}a_{N_{0}}/{\sigma}_{N}$ (since this
term is of order $o({\sigma}_{N}^{-r}\ln{\sigma}_{N}){\sigma}_{N}^{q}$). We
conclude that, up to a term of order $o({\sigma}_{N}^{-r})$, the total
contribution of $t_{j}$ is

$e^{-it_{j}k}\;e^{-k_{N}^{2}/2}\left(\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\sum_{q=2}^{r}\frac{\mathscr{H}_{N,q}(k_{N};t_{j})}{{\sigma}_{N}^{q}}\right)$

where
$\displaystyle\mathscr{H}_{N,q}(x;t_{j})={\mathcal{C}}_{N,q}(x)-\frac{xa_{N_{0}}(i)^{q-2}H_{q-2}(x)\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(q-2)}(t_{j})}{(q-2)!}$
which completes the proof of (7.3).

Next we prove (7.5). On the one hand, by assumption we have Edgeworth
expansions or order $r$, and, on the other hand, we have the expansions from
Theorem 2.1. Therefore, the difference between the two must be
$o({\sigma}_{N}^{-r})$. Since the usual Edgeworth expansions contain no terms
corresponding to nonzero resonant points, applying Lemma 5.1 and (7.3) we
obtain (7.5). ∎

Note that the formulas of Lemma 7.2 together with already proven Proposition
7.1 give the following result.

######  Corollary 7.4.

Suppose that $\mathbb{E}(S_{N})$ is bounded, $S_{N}$ admits the Edgeworth
expansion of order $r-1$, and, either

(a) for some ${\bar{\varepsilon}}\leq 1/(8K)$ we have
$N_{0}=N_{0}(N,t_{j},{\bar{\varepsilon}})=0$ for each each nonzero resonant
point $t_{j}$,

or (b)
$\displaystyle\varphi:=\min_{t\in{\mathcal{R}}}\inf_{n}|\phi_{n}(t)|>0$.

Then

$\sqrt{2\pi}\mathbb{P}(S_{N}=k)$
$=e^{-k_{N}^{2}/2}\left[{\mathcal{E}}_{r}(k_{N})+\sum_{t_{j}\in{\mathcal{R}}}\left(\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}+\frac{ik_{N}C_{1,N,t_{j}}\Phi_{N}(t_{j})}{{\sigma}_{N}^{2}}\right)e^{-it_{j}k}\right]+o(\sigma_{N}^{-r})$

where ${\mathcal{E}}_{r}(\cdot)$ is the Edgeworth polynomial of order $r$
(i.e. the contribution of $t=0$), and we recall that

$iC_{1,N,t_{j}}=-\sum_{n=1}^{N}\frac{{\mathbb{E}}(e^{it_{j}X_{n}}\bar{X}_{n})}{{\mathbb{E}}(e^{it_{j}X_{n}})}.$

###### Proof.

Part (a) holds since under the assumption that $N_{0}=0$ all terms
${\mathcal{H}}_{N,q,j}$ in (7.4) except ${\mathcal{H}}_{N,2,2}$ vanish. Part
(b) holds since in this case the argument proceeds similarly to the proof of
Theorem 2.1 if we set $N_{0}=0$ for any $t_{j}$ (since we only needed $N_{0}$
to obtain a positive lower bound on $|\phi_{n}(t_{j})|$ for
$t_{j}\in{\mathcal{R}}$ and $N_{0}<n\leq N$). ∎

######  Remark 7.5.

Observe that
$\displaystyle{\sigma}_{N}^{-1}\gg|C_{1,N,{t_{j}}}|{\sigma}_{N}^{-2},$ so if
the conditions of the corollary are satisfied but $|\Phi_{N}(t_{j})|\leq
c{\sigma}_{N}^{1-r}$ (possibly along a subsequence), then the leading
correction to the Edgeworth expansion comes from

$e^{-k_{N}^{2}/2}\sum_{t_{j}\in{\mathcal{R}}}\left(\frac{\Phi_{N}(t_{j})}{{\sigma}_{N}}\right).$

Thus Corollary 7.4 strengthens Corollary 7.3 by computing the leading
correction to the Edgeworth expansion when the expansion does not hold.

###  7.3. Proof of Theorem 1.5

We will use the following.

######  Lemma 7.6.

Let $t_{j}$ be a nonzero resonant point, $r>1$ and suppose that $M_{N}\leq
R\ln{\sigma}_{N}$, $R=R(r,K)$ and that
$|{\mathbb{E}}(S_{N})|=O(\ln{\sigma}_{N})$. Then (7.1) holds for all
$0\leq\ell<r$ if and only if

(7.10) $|\Phi_{N}^{(\ell)}(t_{j})|=o\left({\sigma}_{N}^{-r+\ell+1}\right)$

for all $0\leq\ell<r$.

###### Proof.

Let us first assume that (7.1) holds. Recall that

(7.11) $\Phi_{N}(t)=\Phi_{N_{0}}(t)\Phi_{N_{0},N}(t)$

with

(7.12)
$\Phi_{N_{0},N}(t)=\Phi_{N_{0},N}(t_{j})\Phi_{N_{0},N}(h)\Psi_{N_{0},N}(h)$

where $t=t_{j}+h$ and

(7.13)
$\Psi_{N_{0},N}(h)=\exp\left[O(M_{N}(m))\sum_{u=1}^{\infty}(O(1))^{u}h^{u}\right].$

For $\ell=0$ the result reduces to (7.11). For larger $\ell$’s we have

(7.14)
$\Phi_{N}^{(\ell)}(t_{j})=\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})+\sum_{k=0}^{\ell-1}\binom{\ell}{k}\Phi_{N_{0},N}^{(\ell-k)}(t_{j})\Phi_{N_{0}}^{(k)}(t_{j}).$

Fix some $k<\ell$. Then by (7.12),

$\displaystyle\Phi_{N_{0},N}^{(\ell-k)}(t_{j})=\Phi_{N_{0},N}(t_{j})\sum_{u=0}^{\ell-k}\binom{\ell-k}{u}\Phi_{N_{0},N}^{(u)}(0)\Psi_{N_{0},N}^{(\ell-
k-u)}(0)$ $\displaystyle=O(\ln^{\ell-k}{\sigma}_{N})\Phi_{N_{0},N}(t_{j})$

where we have used that
$\bar{S}_{N_{0},N}=S_{N_{0},N}-{\mathbb{E}}(S_{N_{0},N})$ satisfies
$|{\mathbb{E}}[(\bar{S}_{N_{0},N})^{q}]|\leq C_{q}{\sigma}_{N_{0},N}^{2q}$,
(see (3.3)). Therefore

(7.15)
$\Phi_{N_{0},N}^{(\ell-k)}(t_{j})\Phi_{N_{0}}^{(k)}(t_{j})=O(\ln^{\ell-k}{\sigma}_{N})\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(k)}(t_{j}).$

Finally, by (7.1) we have

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(k)}(t_{j})=o({\sigma}_{N}^{k+1-r})$

and so, since $k<\ell$,

$\Phi_{N_{0}}^{(\ell-k)}(t_{j})\Phi_{N_{0}}^{(k)}(t_{j})=o({\sigma}_{N}^{\ell+1-r}).$

This completes the proof that (7.10) holds.

Next, suppose that (7.10) holds for each $0\leq\ell<r$. Let use prove by
induction on $\ell$ that

(7.16)
$|\Phi_{N_{0},N}\Phi_{N_{0}}^{(\ell)}(t_{j})|=o\left({\sigma}_{N}^{-r+\ell+1}\right).$

For $\ell=1$ this follows from (7.14). Now take $\ell>1$ and assume that
(7.16) holds with $k$ in place of $\ell$ for each $k<\ell$. Then by (7.14),
(7.15) and the induction hypothesis we get that

$\Phi_{N}^{(\ell)}(t_{j})=\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})+o({\sigma}_{N}^{\ell+1-r}).$

By assumption we have $\Phi_{N}^{(\ell)}(t_{j})=o({\sigma}_{N}^{\ell+1-r})$
and hence

$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{(\ell)}(t_{j})=o({\sigma}_{N}^{\ell+1-r})$

as claimed. ∎

Theorem 1.5 in the case $M_{N}\leq R\ln\sigma_{N}$ follows now by first
replacing $X_{n}$ with $X_{n}-c_{n}$, where $(c_{n})$ is a bounded sequence of
integers so that ${\mathbb{E}}[S_{N}-C_{N}]=O(1)$, where

(7.17) $C_{N}=\sum_{j=1}^{n}c_{j}$

(see Lemma 3.4 in [10]), and then applying Lemma 7.6 and Proposition 7.1.

It remains to consider the case when $M_{N}(m)\geq{\bar{R}}\ln\sigma_{N}$
where ${\bar{R}}$ is large enough. In that case, by Theorem 1.4, the Edgeworth
expansion of order $r$ hold true, and so, after the reduction to the case when
${\mathbb{E}}(S_{N})$ is bounded, it is enough to show that
$\displaystyle\Phi_{N}^{(\ell)}(t_{j})=o\left(\sigma_{N}^{-r+\ell+1}\right)$
for all $0\leq\ell<r.$ By the arguments of Lemma 7.6 it suffices to show that
for each $0\leq\ell<r$ we have
$\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})=o(\sigma_{N}^{-r})$. To
this end we write

$\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})=\sum_{n_{1},\dots,n_{k}\leq
N_{0};\atop\ell_{1}+\dots+\ell_{k}=\ell}\gamma_{\ell_{1},\dots,\ell_{k}}\left(\prod_{q=1}^{k}\phi_{n_{q}}^{(\ell_{q})}(t_{j})\right)\left[\prod_{n\leq
N,\;n\neq n_{k}}\phi_{n}(t_{j})\right]$

where $\gamma_{\ell_{1},\dots,\ell_{k}}$ are bounded coefficients of
combinatorial nature. Using (3.8) we see that for each $n_{1},\dots,n_{k}$ the
product in the square brackets is at most $Ce^{-c_{0}M_{N}(m)+O(1)}$ for some
$C,c_{0}>0$. Hence

$|\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})|\leq\hat{C}N_{0}^{\ell}\;e^{-cM_{N}(m)}.$

It remains to observe that the definition of $N_{0}$ gives
$M_{N}(m)\geq\hat{\varepsilon}N_{0}.$ Therefore
$\displaystyle|\Phi_{N_{0}}^{(\ell)}(t_{j})\Phi_{N_{0},N}(t_{j})|\leq
C^{*}M_{N}^{\ell}(m)\;e^{-cM_{N}(m)}=o(\sigma_{N}^{-r})$ provided that
$M_{N}\geq{\bar{R}}\ln\sigma_{N}$ for ${\bar{R}}$ large enough. ∎

##  8\. Edgeworth expansions and uniform distribution.

###  8.1. Proof of Theorem 1.6

In view of Proposition 7.1 with $r=2$, it is enough to show that if
$\Phi_{N}(t_{j})=o(\sigma_{N}^{-1})$ then the SLLT implies that

(8.1) $|\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{\prime}(t_{j})|=o(1)$

for any non-zero resonant point $t_{j}$ (note that the equivalence of
conditions (b) and (c) of the theorem follows from Lemma 6.2).

Denote $\displaystyle\Phi_{N;k}(t)=\prod_{l\neq k,l\leq N}\phi_{l}(t)$.

Let us first assume that $\phi_{k}(t_{j})\not=0$ for all $1\leq k\leq N$. Then
$\phi_{k}^{\prime}(t_{j})\Phi_{N;k}(t_{j})=\phi_{k}^{\prime}(t_{j})\Phi_{N}(t_{j})/\phi_{k}(t_{j})$.
Let ${\varepsilon}_{N}=\frac{\ln{\sigma}_{N}}{{\sigma}_{N}}$. If for all
$1\leq k\leq N_{0}$ we have $|\phi_{k}(t_{j})|\geq{\varepsilon}_{N}$ then

$\left|\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{\prime}(t_{j})\right|=\left|\sum_{k=1}^{N_{0}}\phi_{k}^{\prime}(t_{j})\Phi_{N;k}(t_{j})\right|\leq|\Phi_{N}(t_{j})|\sum_{k=1}^{N_{0}}|\phi^{\prime}_{k}(t_{j})/\phi_{k}(t_{j})|$
$\leq C{\varepsilon}_{N}^{-1}N_{0}|\Phi_{N}(t_{j})|\leq
C^{\prime}{\sigma}_{N}|\Phi_{N}(t_{j})|\to 0\text{ as }N\to\infty$

where we have used that $N_{0}=O(\ln V_{N})$. Next suppose there is at least
one $1\leq k\leq N_{0}$ such that $|\phi_{k}(t_{j})|<{\varepsilon}_{N}$. Let
us pick some $k=k_{N}$ with the latter property. Then for any $k\not=k_{N}$,
$1\leq k\leq N_{0}$ we have

$|\phi_{k}^{\prime}(t_{j})\Phi_{N;k}(t_{j})|\leq
C|\phi_{k_{N}}(t_{j})|<C{\varepsilon}_{N}.$

Therefore,

$\left|\sum_{k\not=k_{N},\,1\leq k\leq
N_{0}}\phi_{k}^{\prime}(t_{j})\Phi_{N;k}(t_{j})\right|\leq\frac{C^{\prime}\ln^{2}{\sigma}_{N}}{{\sigma}_{N}}=o(1).$

It follows that

(8.2)
$\Phi_{N_{0},N}(t_{j})\Phi_{N_{0}}^{\prime}(t_{j})=\Phi_{N;k_{N}}(t_{j})\phi_{k_{N}}^{\prime}(t_{j})+o(1).$

Next, in the case when $\phi_{k_{0}}(t_{j})=0$ for some $1\leq k_{0}\leq
N_{0}$, then (8.2) clearly holds true with $k_{N}=k_{0}$ since all the other
terms vanish.

In summary, either (8.1) holds or we have (8.2). In the later case, using
(3.8) we obtain

(8.3) $\left|\mathbb{E}\left(e^{it_{j}S_{N;k_{N}}}\right)\right|\leq
e^{-c_{2}\sum_{s\not=k_{N},1\leq s\leq
N}q_{s}(m)}=e^{-c_{2}M_{N}(m)-q_{k_{N}}(m)}$

where $S_{N;k}=S_{N}-X_{k}$, and $c_{2}>0$ depends only on $K$. Since the SLLT
holds true, $M_{N}$ converges to $\infty$ as $N\to\infty$. Taking into account
that $0\leq q_{k_{N}}(m)\leq 1$ we get that the left hand side of (8.3)
converges to $0$, proving (8.1). ∎

###  8.2. Proof of Theorem 1.8

We start with the proof of part (1). Assume that the LLT holds true in a
superstable way. Let $X_{1}^{\prime},X_{2}^{\prime},...$ be a square
integrable integer-valued independent sequence which differs from
$X_{1},X_{2},...$ by a finite number of elements. Then there is
$n_{0}\in{\mathbb{N}}$ so that $X_{n}=X^{\prime}_{n}$ for any $n>n_{0}$. Set
$\displaystyle S_{N}^{\prime}=\sum_{n=1}^{N}X^{\prime}_{n}$,
$Y=S^{\prime}_{n_{0}}$ and
$Y_{N}=Y{\mathbb{I}}(|Y|<{\sigma}_{N}^{1/2+{\varepsilon}})$, where
${\varepsilon}>0$ is a small constant. By the Markov inequality we have

${\mathbb{P}}(|Y|\geq{\sigma}_{N}^{1/2+{\varepsilon}})={\mathbb{P}}(|Y|^{2}\geq{\sigma}_{N}^{1+2{\varepsilon}})\leq\|Y\|_{L^{2}}^{2}{\sigma}_{N}^{-1-2{\varepsilon}}=o({\sigma}_{N}^{-1}).$

Therefore, for any $k\in{\mathbb{N}}$ and $N>n_{0}$ we have

$\displaystyle{\mathbb{P}}(S^{\prime}_{N}=k)={\mathbb{P}}(S_{N;1,2,...,n_{0}}+Y_{N}=k)+o({\sigma}_{N}^{-1})$
$\displaystyle={\mathbb{E}}[{\mathbb{P}}(S_{N;1,2,...,n_{0}}=k-Y_{N}|X_{1}^{\prime},...,X_{n_{0}}^{\prime})]+o({\sigma}_{N}^{-1})$
$\displaystyle={\mathbb{E}}[P_{N:n_{1},...,n_{0}}(k-Y_{N})]+o({\sigma}_{N}^{-1})$

where $P_{N:n_{1},...,n_{0}}(s)={\mathbb{P}}(S_{N;1,2,...,n_{0}}=s)$ for any
$s\in{\mathbb{Z}}$. Since the LLT holds true in a superstable way, we have,
uniformly in $k$ and the realizations of
$X_{1}^{\prime},...,X_{n_{0}}^{\prime}$ that

$P_{N:n_{1},...,n_{0}}(k-Y_{N})=\frac{e^{-(k-Y_{N}-{\mathbb{E}}(S_{N}))^{2}/(2V_{N})}}{\sqrt{2\pi}{\sigma}_{N}}+o({\sigma}_{N}^{-1}).$

Therefore,

(8.4) ${\mathbb{P}}(S^{\prime}_{N}=k)=$
$\frac{e^{-(k-{\mathbb{E}}(S_{N}))^{2}/2V_{N}}}{\sqrt{2\pi}{\sigma}_{N}}{\mathbb{E}}\big{(}e^{-(k-{\mathbb{E}}(S_{N}))Y_{N}/V_{N}+Y_{N}^{2}/(2V_{N})}\big{)}+o({\sigma}_{N}^{-1}).$

Next, since $|Y_{N}|\leq{\sigma}_{N}^{1/2+{\varepsilon}}$ we have that
$\|Y_{N}^{2}/(2V_{N})\|_{L^{\infty}}\leq{\sigma}_{N}^{2{\varepsilon}-1}$, and
so when ${\varepsilon}<1/2$ we have $\|Y_{N}^{2}/2V_{N}\|_{L^{\infty}}=o(1)$.
Recall that $k_{N}=(k-{\mathbb{E}}(S_{N}))/{\sigma}_{N}$. Suppose first that
$|k_{N}|\geq{\sigma}_{N}^{{\varepsilon}}$ with ${\varepsilon}<1/4.$

Since

$\big{|}(k-{\mathbb{E}}(S_{N}))Y_{N}/V_{N}\big{|}\leq|k_{N}|{\sigma}_{N}^{{\varepsilon}-\frac{1}{2}},$

we get that the RHS of (8.4) is $o({\sigma}_{N}^{-1})$ (uniformly in such
$k$’s).

On the other hand, if $|k_{N}|<{\sigma}_{N}^{{\varepsilon}}$ then

${\mathbb{E}}\big{(}e^{-(k-{\mathbb{E}}(S_{N}))Y_{N}/V_{N}+Y_{N}^{2}/2V_{N}}\big{)}=1+o(1)$

(uniformly in that range of $k$’s).

We conclude that, uniformly in $k$, we have

${\mathbb{P}}(S^{\prime}_{N}=k)=\frac{e^{-(k-{\mathbb{E}}(S_{N}))^{2}/(2V_{N})}}{\sqrt{2\pi}{\sigma}_{N}}+o({\sigma}_{N}^{-1}).$

Lastly, since
$\displaystyle\sup_{N}|{\mathbb{E}}(S_{N})-{\mathbb{E}}(S_{N}^{\prime})|\\!\\!<\\!\infty$
and
$\displaystyle\sup_{N}|\text{Var}(S_{N})-\text{Var}(S_{N}^{\prime})|\\!<\\!\\!\infty,$

${\mathbb{P}}(S^{\prime}_{N}=k)=\frac{e^{-(k-{\mathbb{E}}(S^{\prime}_{N}))^{2}/(2V^{\prime}_{N})}}{\sqrt{2\pi}{\sigma}^{\prime}_{N}}+o(1/{\sigma}_{N}^{\prime})$

where $V_{N}^{\prime}=\text{Var}(S_{N}^{\prime})$ and
${\sigma}_{N}^{\prime}=\sqrt{V_{N}^{\prime}}$.

Conversely, if the SLLT holds then $M_{N}(h)\to\infty$ for each $h\geq 2$. Now
if $t$ is a nonzero resonant point with denominator $h$ then (3.8) gives

$|\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)|\leq
Ce^{-cM_{N}(h)+\bar{C}{\bar{s}}},\,\,C,\bar{C}>0$

for any choice of $j_{1}^{N},...,j_{s_{N}}^{N}$ and $\bar{s}$ with
$s_{N}\leq\bar{s}$. Since the RHS tends to 0 as $N\to\infty$, $\\{X_{n}\\}\in
EeSS(1)$ completing the proof of part (1).

For part (2) we only need to show that (a) is equivalent to (b) as the
equivalence of (b) and (c) comes from Lemma 6.2. By replacing again $X_{n}$
with $X_{n}-c_{n}$ it is enough to prove the equivalency in the case when
${\mathbb{E}}(S_{N})=O(1)$. The proof that (a) and (b) are equivalent consists
of two parts. The first part is the following statement whose proof is a
straightforward adaptation of the proof of Theorem 1.5 and is therefore
omitted.

######  Proposition 8.1.

$\\{X_{n}\\}\in SsEe(r)$ if and only if for each ${\bar{s}}$, each sequence
$j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$ with $s_{N}\leq{\bar{s}}$, each
$\ell<r$ and each $t\in{\mathcal{R}}$ we have

(8.5)
$\Phi^{(\ell)}_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma_{N}^{\ell+1-r}).$

Note that the above proposition shows that the condition
$\displaystyle\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma^{1-r}_{N})$
is necessary.

The second part of the argument is to show that if

$\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma^{1-r}_{N})$

holds for every finite modification of $S_{N}$ with $s_{N}\leq{\bar{s}}+\ell$
(uniformly) then (8.5) holds for every modifications with $s_{N}\leq{\bar{s}}$
so that the condition
$\displaystyle\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma^{1-r}_{N})$
is also sufficient.

To this end we introduce some notation. Fix a nonzero resonant point
$t=\frac{2\pi l}{m}.$ Let $\check{\Phi}_{N}$ be the characteristic function of
the sum $\check{S}_{N}$ of all $X_{n}$’s such that $1\leq n\leq N$,
$n\not\in\\{j_{1}^{N},j_{2}^{N},...,j_{s_{N}}^{N}\\}$ and
$q_{n}(m)\geq\bar{\epsilon}$. Let $\check{N}$ be the number of terms in
$\check{S}_{N}.$ Denote
$\tilde{S}_{N}=S_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}-\check{S}_{N}$
and let $\tilde{\Phi}_{N}(t)$ be the characteristic function of
$\tilde{S}_{N}.$ Similarly to the proof of Theorem 1.5 it suffices to show
that for each $\ell<r$

$\left|\check{\Phi}_{N}^{(\ell)}\tilde{\Phi}_{N}(t)\right|=o(\sigma_{N}^{1+\ell-r})$

and, moreover, we can assume that $M_{N}(m)\leq{\bar{R}}\ln\sigma_{N}$ so that
$\check{N}=O(\ln\sigma_{N}).$ We have (cf. (4.21)) ,

$\check{\Phi}_{N}^{(\ell)}\tilde{\Phi}_{N}(t)=\sum_{\begin{subarray}{c}n_{1},\dots,n_{k};\;\\\
\ell_{1}+\dots+\ell_{k}=\ell\end{subarray}}\prod_{w=1}^{k}\gamma_{\ell_{1},\dots,\ell_{k}}\phi_{n_{w}}^{(\ell_{w})}(t_{j})\prod_{n\not\in\\{n_{1},n_{2}\dots,n_{k},j_{1}^{N},j_{2}^{N}\dots,j_{s_{N}}^{N}\\}}\phi_{n}(t_{j})$

where the summation is over all tuples $n_{1},n_{2},\dots,n_{k}$ such that
$q_{n_{w}}(m)\geq\bar{\epsilon}$. Note that the absolute value of each term in
the above sum is bounded by
$C|\Phi_{N;n_{1},\dots,n_{k},j_{1}\dots,j_{s_{N}}^{N}}(t_{j})|=o(\sigma_{N}^{1-r}).$
It follows that the whole sum is

$o\left(\sigma_{N}^{1-r}\check{N}^{\ell}\right)=o\left(\sigma_{N}^{1-r}\ln^{\ell}\sigma_{N}\right)$

completing the proof. ∎

######  Remark 8.2.

Lemma 6.2 and Theorem 1.5 show that the convergence to uniform distribution on
any factor $\mathbb{Z}/h\mathbb{Z}$ with the speed $o({\sigma}_{N}^{1-r})$ is
necessary for Edgeworth expansion of order $r.$ This is quite intuitive.
Indeed calling $\mathscr{E}_{r}$ the Edgeworth function of order $r$, (i.e.
the contribution from zero), then it is a standard result from numerical
integration (see, for instance, [9, Lemma A.2]) that for each $s\in\mathbb{N}$
and each $j\in\mathbb{Z}$

$\sum_{k\in\mathbb{Z}}h\mathscr{E}_{r}\left(\frac{j+hk}{\sqrt{{\sigma}_{N}}}\right)=\int_{-\infty}^{\infty}\mathscr{E}_{r}(x)dx+o\left({\sigma}_{N}^{-s}\right)=1+o\left({\sigma}_{N}^{-s}\right)$

where in the last inequality we have used that the non-constant Hermite
polynomials have zero mean with respect to the standard normal law (since they
are orthogonal to the constant functions). However, using this result to show
that

$\displaystyle\sum_{k\in\mathbb{Z}}\mathbb{P}(S_{N}=j+kh)=\frac{1}{h}+o\left({\sigma}_{N}^{1-r}\right)$

requires a good control on large values of $k.$ While it appears possible to
obtain such control using the large deviations theory it seems simpler to
estimate the convergence rate towards uniform distribution from our
generalized Edgeworth expansion.

##  9\. Second order expansions

In this section we will compute the polynomials in the general expansions in
the case $r=2$.

First, let us introduce some notations which depend on a resonant point
$t_{j}.$ Let $t_{j}=2\pi l_{j}/m_{j}$ be a nonzero resonant point such that
$M_{N}(m_{j})\leq R(2,K)\ln V_{N}$ where $R(2,K)$ is specified in Remark 3.6.
Let $\check{\Phi}_{j,N}$ be the characteristic function of the sum
$\check{S}_{j,N}$ of all $X_{n}$’s such that $1\leq n\leq N$ and
$q_{n}(m_{j})\geq\bar{\epsilon}=\frac{1}{8K}$. Note that $\check{S}_{j,N}$ was
previously denoted by $S_{N_{0}}$. Let $\tilde{S}_{N,j}=S_{N}-\check{S}_{N,j}$
and denote by $\tilde{\Phi}_{N,j}$ its characteristic function. (In previous
sections we denoted the same expression by $S_{N_{0},N}$, but here we want to
emphasize the dependence on $t_{j}$.) Let $\gamma_{N,j}$ be the ratio between
the third moment of $\tilde{S}_{N,j}-{\mathbb{E}}(\tilde{S}_{N,j})$ and its
variance. Recall that by (3.3) $|\gamma_{N,j}|\leq C$ for some $C$. Also, let
$C_{1,N,t_{j}}$ be given by (4.15), with the indexes rearranged so that the
$n$’s with $q_{n}(m)\geq\bar{\varepsilon}$ are the first $N_{0}$ ones
($C_{1,N,t_{j}}$ is at most of order $M_{N}(m)=O(\ln V_{N})$). For the sake of
convenience, when either $t_{j}=0$ or $M_{N}(m_{j})\geq R(2,K)\ln V_{N}$ we
set $C_{1,N,t_{j}}=0$, $\tilde{S}_{N,j}=S_{N}$ and $\check{S}_{N,j}=0$. In
this case $\tilde{\Phi}_{N,j}=\tilde{\Phi}_{N}$ and $\check{\Phi}_{N,j}\equiv
1$. Also denote $k_{N}=(k-{\mathbb{E}}(S_{N}))/{\sigma}_{N},$
$\bar{S}_{N}=S_{N}-{\mathbb{E}}(S_{N})$, and
$\gamma_{N}={\mathbb{E}}(\bar{S}_{N}^{3})/V_{N}$, ($\gamma_{N}$ is bounded).

######  Proposition 9.1.

Uniformly in $k$, we have

(9.1)
$\displaystyle\sqrt{2\pi}{\mathbb{P}}(S_{N}=k)=\Big{(}1+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})\Big{)}{\sigma}_{N}^{-1}e^{-k_{N}^{2}/2}$
$\displaystyle-{\sigma}_{N}^{-2}e^{-k_{N}^{2}/2}\left(\gamma_{N}k_{N}^{3}/6+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\tilde{\Phi}_{N,j}(t_{j})P_{N,j}(k_{N})\right)$
$\displaystyle+o(\sigma_{N}^{-2})$

where

$P_{N,j}(x)=\big{(}\check{\Phi}_{N,j}(t_{j})(iC_{1,N,t_{j}}-{\mathbb{E}}(\check{S}_{N,j}))+i\check{\Phi}_{N,j}^{\prime}(t_{j})\big{)}x+\check{\Phi}_{N,j}(t_{j})\gamma_{N,j}x^{3}/6.$

###### Proof.

Let $t_{j}=\frac{2\pi l}{m}$ be a resonant point with $M_{N}(m)\leq R(2,K)\ln
V_{N}.$ Recall that $\textbf{C}_{j}(k)$ are given by (4.22). First, in order
to compute the term corresponding to ${\sigma}_{N_{0},N}^{-2}$ we need only to
consider the case $s\leq 1$ in (4.23). Using (3.2) we end up with the
following contribution of the interval containing $t_{j}$,

$\sqrt{(2\pi)^{-1}}e^{-it_{j}k}\tilde{\Phi}_{N,j}(t_{j}){\sigma}_{N_{0},N}^{-1}\Bigg{(}\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb{E}}[\tilde{S}_{N,j}])/{\sigma}_{N,j}}e^{-h^{2}/2}dh$
$+{\sigma}_{N,j}^{-1}\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb{E}}[\tilde{S}_{N,j}])/{\sigma}_{N,j}}\left(\frac{ih^{3}}{6}{\mathbb{E}}\left[\big{(}\tilde{S}_{N,j}-{\mathbb{E}}(\tilde{S}_{N,j})\big{)}^{3}\right]{\sigma}_{N,j}^{-3}\right)dh$
$+{\sigma}_{N,j}^{-1}(C_{1,N}\check{\Phi}_{N,j}(t_{j})+\check{\Phi}_{N,j}^{\prime}(t_{j}))\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb{E}}(\tilde{S}_{N,j}))/{\sigma}_{N,j}}he^{-h^{2}/2}dh\Bigg{)}$
$=e^{-it_{j}k}\tilde{\Phi}_{N,j}(t_{j})\sqrt{(2\pi)^{-1}}e^{-k_{N,j}^{2}/2}{\sigma}_{N,j}^{-1}\Big{(}\check{\Phi}_{N,j}(t_{j})+i\big{(}C_{1,N,t_{j}}\check{\Phi}_{N,j}(t_{j})+\check{\Phi}_{N,j}^{\prime}(t_{j})\big{)}$
$\times
k_{N,j}{\sigma}_{N,j}^{-1}+\check{\Phi}_{N,j}(t_{j})(k_{N,j}^{3}-3k_{N,j})\gamma_{N,j}{\sigma}_{N,j}^{-1}/6\Big{)}$

where ${\sigma}_{N,j}=\sqrt{V(\tilde{S}_{N,j})}$,
$k_{N,j}=(k-{\mathbb{E}}(\tilde{S}_{N,j}))/{\sigma}_{N,j}$ and
$\gamma_{N,j}=\frac{{\mathbb{E}}[(\tilde{S}_{N,j}-{\mathbb{E}}(\tilde{S}_{N,j}))^{3}]}{{\sigma}_{N,j}^{2}}$
(which is uniformly bounded).

As before we shall only consider the case where $|k_{N}|\leq
V_{N}^{\varepsilon}$ with ${\varepsilon}=0.01$ since otherwise both the LHS
and the RHS (9.1) are $O({\sigma}_{N}^{-r})$ for all $r.$ Then, the last
display can be rewritten as $I+I\\!\\!I$ where

(9.2)
$I=\frac{e^{-it_{j}k}}{\sqrt{2\pi}{\sigma}_{N,j}}\;e^{-k_{N,j}^{2}/2}\;\Phi_{N}(t_{j});$
$I\\!\\!I=\frac{e^{-it_{j}k}}{\sqrt{2\pi}{\sigma}^{2}_{N,j}}\;e^{-k_{N,j}^{2}/2}\times$
$\left[\Phi_{N}(t_{j})\left(iC_{1,N,t_{j}}k_{N,j}+\frac{\gamma_{N,j}}{6}\left(k_{N,j}^{3}-3k_{N,j}\right)\right)+i\check{\Phi}_{N,j}^{\prime}(t_{j})\tilde{\Phi}_{N,j}(t_{j})k_{N,j}\right].$

In the region $|k_{N}|\leq V_{N}^{\varepsilon}$ we have

$I=\frac{e^{-it_{j}k}}{\sqrt{2\pi}{\sigma}_{N}}\;e^{-k_{N}^{2}/2}\;\left[1-q_{N,j}k_{N}\right]\Phi_{N}(t_{j})+o\left({\sigma}_{N}^{-2}\right)$

where

$q_{N,j}={\mathbb{E}}(\check{S}_{N,j})/{\sigma}_{N,j}={\mathbb{E}}(\check{S}_{N,j})/{\sigma}_{N}+O(\ln{\sigma}_{N}/{\sigma}_{N}^{3})=O(\ln{\sigma}_{N}/{\sigma}_{N})$

while

$I\\!\\!I=\frac{e^{-it_{j}k}}{\sqrt{2\pi}{\sigma}^{2}_{N}}\;e^{-k_{N}^{2}/2}\times$
$\left[\Phi_{N}(t_{j})\left(iC_{1,N,t_{j}}k_{N}+\frac{{\gamma}_{N,j}\left(k_{N}^{3}-3k_{N}\right)}{6}\right)+i\check{\Phi}_{N,j}^{\prime}(t_{j})\tilde{\Phi}_{N,j}(t_{j})k_{N}\right]$
$+o\left({\sigma}_{N}^{-2}\right).$

This yields (9.1) with ${\mathcal{R}}_{N}$ in place of ${\mathcal{R}}$, where
${\mathcal{R}}_{N}$ is the set on nonzero resonant points $t_{j}=2\pi l/m$
such that $M_{N}(m)\leq R(2,K)\ln V_{N}$. Next, (3.8) shows that if
$M_{N}(m)\geq R(2,k)\ln V_{N}$ then

$\sup_{t\in I_{j}}|\Phi_{N}(t)|\leq e^{-c_{0}M_{N}(m)}=o({\sigma}_{N}^{-2})$

and so the contribution of $I_{j}$ to the right hand side of (9.1) is
$o({\sigma}_{N}^{-2}).$ Finally, the contribution coming from $t_{j}=0$ is

$e^{-k_{N}^{2}/2}\left({\sigma}_{N}^{-1}+{\sigma}_{N}^{-2}\gamma_{N}^{3}k_{N}^{3}/6\right)$

and the proof of the proposition is complete. ∎

######  Remark 9.2.

Suppose that $M_{N}(m)\geq R(2,K)\ln V_{N}$ and let $N_{0}$ is the number of
$n$’s between $1$ to $N$ so that $q_{n}(m)\geq\frac{1}{8K}$. Then using (3.8)
we also have

$|\check{\Phi}^{\prime}_{N,j}(t_{j})\tilde{\Phi}_{N,j}(t_{j})|\leq\sum_{n\in{\mathcal{B}}_{{\bar{\varepsilon}}}(m)}|{\mathbb{E}}[X_{n}e^{it_{j}X_{n}}]|\cdot|\Phi_{N;n}(t_{j})|$
$\leq CN_{0}(N,t_{j},{\bar{\varepsilon}})e^{-c_{0}M_{N}(m)}\leq
C^{\prime}M_{N}(m)e^{-c_{0}M_{N}(m)},$

where

${\mathcal{B}}_{N,{\bar{\varepsilon}}}(m)=\\{1\leq n\leq
N:q_{n}(m)>{\bar{\varepsilon}}\\}.$

Since $M_{N}(m)\geq R(2,K)\ln V_{N}$, for any $0<c_{1}<c_{0}$, when $N$ is
large enough we have

$M_{N}(m)e^{-c_{0}M_{N}(m)}\leq C_{1}e^{-c_{1}M_{N}(m)}=o({\sigma}_{N}^{-2}).$

Similarly,
$|{\mathbb{E}}(\check{S}_{N,j})\Phi_{N}(t_{j})|=o({\sigma}_{N}^{-2})$ and

$C_{1,N,t_{j}}\Phi_{N}(t_{j})=O(M_{N}(m))\Phi_{N}(t_{j})=o({\sigma}_{N}^{-2}).$

Therefore we get (9.1) when $\tilde{S}_{N,j}$ and $\check{S}_{N,j}$ are
defined in the same way as in the case $M_{N}(m)\leq R(2,K)\ln V_{N}$.

Under additional assumptions the order 2 expansion can be simplified.

######  Corollary 9.3.

If $S_{N}$ satisfies SLLT then

$\sqrt{2\pi}{\mathbb{P}}(S_{N}=k)=\frac{e^{-k_{N}^{2}/2}}{{\sigma}_{N}}\left(1+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})-\frac{\gamma_{N}k_{N}^{3}}{6{\sigma}_{N}}\right)+o(\sigma_{N}^{-2}).$

###### Proof.

The estimates of §8.1 together with (3.8) show that if $S_{N}$ satisfies the
SLLT then for all $j$

$(1+M_{N}(m))\tilde{\Phi}_{N,j}(t_{j})\check{\Phi}_{N,j}(t_{j})=o(1)\text{ and
}\tilde{\Phi}_{N,j}(t_{j})\check{\Phi}^{\prime}_{N,j}(t_{j})=o(1).$

Thus all terms in the second line of (9.1) except the first one make a
negligible contribution, and so they could be omitted. ∎

Next, assume that $S_{N}$ satisfies the LLT but not SLLT. According to
Proposition 1.2, in this case there exists $m$ such that $M_{N}(m)$ is bounded
and for $k=1,\dots,m-1$ there exists $n=n(k)$ such that $\phi_{n}(k/m)=0$. Let
${\mathcal{R}}_{s}$ denote the set of nonzero resonant points
$t_{j}=\frac{2\pi k}{m}$ so that $M_{N}(m)$ is bounded and
$\phi_{\ell_{j}}(t_{j})=0$ for unique $\ell_{j}.$

######  Corollary 9.4.

Uniformly in $k$, we have

$\displaystyle\sqrt{2\pi}{\mathbb{P}}(S_{N}=k)=\Big{(}1+\sum_{t_{j}\in{\mathcal{R}}}e^{-it_{j}k}\Phi_{N}(t_{j})\Big{)}{\sigma}_{N}^{-1}e^{-k_{N}^{2}/2}$
$\displaystyle-{\sigma}_{N}^{-2}e^{-k_{N}^{2}/2}\left(\gamma_{N}k_{N}^{3}/6+\sum_{t_{j}\in{\mathcal{R}}_{s}}ie^{-it_{j}k}\Phi_{N;\ell_{j}}(t_{j})\phi_{\ell_{j}}^{\prime}(t_{j})k_{N}\right)+o(\sigma_{N}^{-2}).$

###### Proof.

As in the proof of Corollary 9.3 we see that the contribution of the terms
with $k/m$ with $M_{N}(m)\to\infty$ is negligible. Next, for terms in
${\mathcal{R}}_{s}$ the only non-zero term in the second line in (9.1)
corresponds to $\Phi_{N;\ell_{j}}(t_{j})\phi_{\ell_{j}}^{\prime}(t_{j})$ while
for the resonant points such that $\phi_{\ell}(t_{j})=0$ for two different
$\ell$s all terms vanish. ∎

##  10\. Examples.

######  Example 10.1.

Suppose $X_{n}$ are iid integer valued with step $h>1.$ That is there is
$s\in{\mathbb{Z}}$ such that ${\mathbb{P}}(X_{n}\in s+h{\mathbb{Z}})=1$ and
$h$ is the smallest number with this property. In this case [18, Theorem
4.5.4] (see also [12, Theorem 5]) shows that there are polynomials $P_{b}$
such that

(10.1)
${\mathbb{P}}(S_{N}=k)=\sum_{b=1}^{r}\frac{P_{b}((k-{\mathbb{E}}[S_{N}])/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}\left(\frac{k-{\mathbb{E}}(S_{N})}{\sigma_{N}}\right)+o(\sigma_{N}^{-r})$

for all $k\in sN+h{\mathbb{Z}}.$ Then

$\sum_{a=0}^{h-1}\sum_{b=1}^{r}e^{2\pi
ia(k-sN)/h}\frac{P_{b}((k-{\mathbb{E}}[S_{N}])/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}((k-{\mathbb{E}}(S_{N}))/\sigma_{N})$

provides $\displaystyle o(\sigma_{N}^{-r})$ approximation to
${\mathbb{P}}(S_{N}=k)$ which is valid for all $k\in{\mathbb{Z}}.$

Next let ${\bar{S}}_{N}=X_{0}+S_{N}$ where $X_{0}$ is bounded and arithmetic
with step 1. Then using the identity

(10.2) ${\mathbb{P}}({\bar{S}}_{N}=k)=\sum_{u\equiv k-sN\text{ mod
}h}{\mathbb{P}}(X_{0}=u){\mathbb{P}}(S_{N}=k-u),$

invoking (10.1) and expanding
$\displaystyle\mathfrak{g}\left(\frac{k-u-{\mathbb{E}}(S_{N})}{\sigma_{N}}\right)$
in the Taylor series about $\frac{k-{{\mathbb{E}}(S_{N})}}{\sigma_{N}}$ we
conclude that there are polynomials $P_{b,j}$ such that we have for $k\in
j+h{\mathbb{Z}}$,

${\mathbb{P}}({\bar{S}}_{N}=k)=\sum_{b=1}^{r}\frac{P_{b,j}((k-{\mathbb{E}}[S_{N}])/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}\left(\frac{k-{\mathbb{E}}(S_{N})}{\sigma_{N}}\right)+o(\sigma_{N}^{-r}).$

Again

$\sum_{a=0}^{h-1}\sum_{j=0}^{h-1}e^{2\pi
ia(k-j)/h}\sum_{b=1}^{r}\frac{P_{b,j}((k-{\mathbb{E}}[S_{N}])/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}\left(\frac{k-{\mathbb{E}}(S_{N})}{\sigma_{N}}\right)$

provides the oscillatory expansion valid for all integers.

######  Example 10.2.

Our next example is a small variation of the previous one. Fix a positive
integer $m.$ Let $X^{\prime}$ be a random variable such that $X^{\prime}$ mod
$m$ is uniformly distributed. Then its characteristic function satisfies
$\phi_{X^{\prime}}(\frac{2\pi a}{m})=0$ for $a=1,\dots,m-1.$ We also assume
that $\phi_{X^{\prime}}^{\prime}(\frac{2\pi a}{m})\neq 0$ for $a$ as above
(for example one can suppose that $X^{\prime}$ takes the values $Lm$,
$1,2,\dots,m-1$ with equal probabilities where $L$ is a large integer). Let
$X^{\prime\prime}$ take values in $m{\mathbb{Z}}$ and have zero mean. We also
assume that $X^{\prime\prime}$ does not take values at $m_{0}{\mathbb{Z}}$ for
a larger $m_{0}$. Then $q(X^{\prime\prime},m_{0})>0$ for any $m_{0}\not=m$.
Fix $r\in\mathbb{N}$ and let

$X_{n}=\begin{cases}X^{\prime}&n\leq r,\\\ X^{\prime\prime}&n>r.\end{cases}$

Then $M_{N}(m_{0})$ grows linearly fast in $N$ if $m_{0}\not=m$ and $M_{N}(m)$
is bounded in $N$. We claim that $S_{N}$ admits the Edgeworth expansion of
order $r$ but does not admit Edgeworth expansion of order $r+1.$ The first
statement holds due to Theorem 1.5, since $\Phi^{(\ell)}_{N}(\frac{2\pi
a}{m})=0$ for each $a\in{\mathbb{Z}}$ and each $\ell<r.$ On the other hand,
since $\Phi_{N}^{(\ell)}(\frac{2\pi a}{m})=0$ for any $\ell<r$, using Lemma
7.6 we see that the conditions of Lemma 7.2 are satisfied with $r+1$ in place
of $r$. Moreover, with $t_{j}=2\pi a/m$, $a\not=0$ we have
${\mathcal{H}}_{N,r+1,s}(x,t_{j})\equiv 0$ for any $q\leq r+1$ and $s=2,3,4$
while ${\mathcal{H}}_{N,q,w}(x,t_{j})\equiv 0$ for any $q\leq r$ and
$w=1,2,3,4$. Furthermore, when $N\geq r$ we have

$\displaystyle{\mathcal{H}}_{N,r+1,1}(x;t_{j})=\frac{i^{r}H_{r}(x)\big{(}\phi_{X^{\prime\prime}}(2\pi
a/m)\big{)}^{N-r}\Phi_{r}^{(r)}(2\pi a/m)}{r!}$
$\displaystyle=(i)^{r}H_{r}(x)\big{(}\Phi_{X^{\prime}}^{\prime}(2\pi
a/m)\big{)}^{r}.$

We conclude that

${\mathbb{P}}(S_{N}=k)$
$=\frac{e^{-k_{N}^{2}/2}}{\sqrt{2\pi}}\left[{\mathcal{E}}_{r+1}(k_{N})+\frac{i^{r}}{{\sigma}_{N}^{r+1}}\sum_{a=1}^{m-1}e^{-2\pi
iak/m}\left(\phi_{X^{\prime}}^{\prime}\left(\frac{2\pi
a}{m}\right)\right)^{r}H_{r}(k_{N})\right]$ $+o({\sigma}_{N}^{-r-1})$

where ${\mathcal{E}}_{r+1}$ the Edgeworth polynomial (i.e. the contribution of
0) and $H_{r}(x)$ is the Hermite polynomial.

Observe that since the uniform distribution on ${\mathbb{Z}}/m{\mathbb{Z}}$ is
shift invariant, $S_{N}$ are uniformly distributed mod $m$ for all
$N\in{\mathbb{N}}.$ This shows that for $r\geq 1$, one can not characterize
Edgeworth expansions just in term of the distributions of $S_{N}$ mod $m$, so
the additional assumptions in Theorems 1.6 and 1.8 are necessary.

Next, consider a more general case where for each $n$, $X_{n}$ equals in law
to either $X^{\prime}$ or $X^{\prime\prime}$, however, now we assume that
$X^{\prime}$ appears infinitely often. In this case $S_{N}$ obeys Edgeworth
expansions of all orders since for large $N$, $\Phi_{N}(t)$ has zeroes of
order greater $N$ at all points of the form $\frac{2\pi a}{m},$
$a=1,\dots,m-1.$ In fact, the Edgeworth expansions hold in the superstable way
since removing a finite number of terms does not make the order of zero to
fall below $r.$

######  Example 10.3.

Let $\mathfrak{p}_{n}=\min(1,\frac{\theta}{n})$ and let $X_{n}$ take value $0$
with probability $\mathfrak{p}_{n}$ and values $\pm 1$ with probability
$\frac{1-\mathfrak{p}_{n}}{2}.$ In this example the only non-zero resonant
point is $\pi=2\pi\times\frac{1}{2}.$ Then for small $\theta$ the
contributions of $P_{1,b,N}$ (the only non-zero $a$ is $1$) are significant
and as a result $S_{N}$ does not admit the ordinary Edgeworth expansion.
Increasing $\theta$ we can make $S_{N}$ to admit Edgeworth expansions of
higher and higher orders. Namely we get that for large $n,$
$\displaystyle\phi_{n}(\pi)=\frac{2\theta}{n}-1.$ Accordingly

$\ln(-\phi_{n}(\pi))=-\frac{2\theta}{n}+O\left(\frac{1}{n^{2}}\right).$

Now the asymptotic relation

$\sum_{n=1}^{N}\frac{1}{n}=\ln N+{\mathfrak{c}}+O\left(\frac{1}{N}\right),$

where ${\mathfrak{c}}$ is the Euler-Mascheroni constant, implies that that
there is a constant $\Gamma(\theta)$ such that

$\Phi_{N}(\pi)=\frac{(-1)^{N}e^{\Gamma(\theta)}}{N^{2\theta}}\left(1+O(1/N)\right).$

Therefore $S_{N}$ admits the Edgeworth expansions of order $r$ iff
$\displaystyle\theta>\frac{r-1}{4}.$ Moreover, if
$\displaystyle\theta\in\left(\frac{r-2}{4},\frac{r-1}{4}\right],$ then
Corollary 7.4 shows that

$\mathbb{P}(S_{N}=k)=\frac{e^{-k_{N}^{2}/2}}{\sqrt{2\pi}}\left[{\mathcal{E}}_{r}(k_{N})+\frac{(-1)^{N+k}e^{\Gamma(\theta)}}{N^{2\theta+(1/2)}}+O\left(\frac{1}{N^{2\theta+1}}\right)\right]$

where ${\mathcal{E}}_{r}$ is the Edgeworth polynomial of order $r.$ In
particular if $\theta\in(0,1/4)$ then using that

(10.3) $V_{N}=N+O(\ln N)=N\left(1+O\left(\frac{\ln N}{N}\right)\right)$

and hence

(10.4) $\sigma_{N}=\sqrt{N}\left(1+O\left(\frac{\ln N}{N}\right)\right)$

we conclude that

$\mathbb{P}(S_{N}=k)=\frac{e^{-k^{2}/(2N)}}{\sqrt{2\pi}}\left[\frac{1}{\sqrt{N}}+\frac{(-1)^{N+k}e^{\Gamma(\theta)}}{N^{2\theta+(1/2)}}+O\left(\frac{1}{N^{2\theta+1}}\right)\right].$

Next, take $\mathfrak{p}_{n}=\min\left(1,\frac{\theta}{n^{2}}\right)$. Then
the SLLT does not hold, since the Prokhorov condition fails. Instead we have
(6.1) with ${\mathcal{R}}=\\{\pi\\}$. Namely, uniformly in $k$ we have

$\sqrt{2}\pi{\mathbb{P}}(S_{N}=k)=\left(1+(-1)^{k}\prod_{u=1}^{N}\left(2\mathfrak{p}_{u}-1\right)\right){\sigma}_{N}^{-1}e^{-k^{2}/2V_{N}}+o(\sigma_{N}^{-1}).$

Next, $\mathfrak{p}_{u}$ is summable and moreover

$\prod_{u=1}^{N}(2\mathfrak{p}_{u}-1)=(-1)^{N}U(1+O(1/N))$

where $\displaystyle U=\prod_{n=1}^{\infty}(1-2\mathfrak{p}_{u})$. We conclude
that

(10.5)
$\sqrt{2}\pi{\mathbb{P}}(S_{N}=k)=\left(1+(-1)^{k+N}U\right){\sigma}_{N}^{-1}e^{-k^{2}/2V_{N}}+O\left(\sigma_{N}^{-2}\right)$

uniformly in $k$. In this case the usual LLT holds true if and only if $U=0$
in agreement with Proposition 1.2.

In fact, in this case we have a faster rate of convergence. To see this we
consider expansions of order $2$ for $\mathfrak{p}_{n}$ as above. We observe
that $q_{m}(2)=\mathfrak{p}_{n}$ for large $n.$ Thus

$|{\mathbb{E}}(e^{\pi iX_{n}})|=1-2\mathfrak{p}_{n}$

and so $|{\mathbb{E}}(e^{\pi X_{n}})|\geq\frac{1}{2}$ when $n\geq N_{\theta}$
for some minimal $N_{\theta}$. Therefore, we can take $N_{0}=N_{\theta}$. Note
also that we have $Y_{n}=X_{n}\text{ mod }2-1$. We conclude that for $n>N_{0}$
we have

$a_{n}=a_{n,j}=\frac{{\mathbb{E}}[((-1)^{Y_{n}}-1)X_{n}]}{{\mathbb{E}}[(-1)^{Y_{n}}]}=0$

and so the term $C_{1,N}$ vanishes. Next, we observe that

$\gamma_{N,j}=\frac{\sum_{n=N_{0}+1}^{N}{\mathbb{E}}(X_{n}^{3})}{\sum_{n=N_{0}+1}^{N}(1-\mathfrak{p}_{n})}=0.$

Finally, we note that ${\mathbb{E}}[(-1)^{X_{n}}X_{n}]=0$, and hence
$\Phi_{N_{0}}^{\prime}(\pi)=0$. Therefore, the second term in (9.1) vanishes
and we have

$\sqrt{2}\pi{\mathbb{P}}(S_{N}=k)=\left(1+(-1)^{k+N}U\right){\sigma}_{N}^{-1}e^{-k^{2}/(2V_{N})}+O\left(\sigma_{N}^{-3}\right).$

Taking into account (10.3) and (10.4) we obtain

$\sqrt{2}\pi{\mathbb{P}}(S_{N}=k)=\frac{1+(-1)^{k+N}U}{\sqrt{N}}\;e^{-k^{2}/(2N)}+O\left(N^{-3/2}\right).$

In particular, (10.5) holds with the stronger rate
$O\left(\sigma_{N}^{-3}\right)$.

######  Example 10.4.

The last example exhibited significant simplifications. Namely, there was only
one resonant point, and, in addition, the second term vanished due to the
symmetry. We now show how a similar analysis could be performed when the above
simplifications are not present. Let us assume that $X_{n}$ takes the values
$-1,0$ and $3$ with probabilities $a_{n},b_{n}$ and $c_{n}$ so that
$a_{n}+b_{n}+c_{n}=1$. Let us also assume that $b_{n}<\frac{1}{8}$ and that
$a_{n},c_{n}\geq\rho>0$ for some constant $\rho$. Then

$V(X_{n})=9(c_{n}-c_{n}^{2})+6a_{n}c_{n}+(a_{n}-a_{n}^{2})\geq 6\rho^{2}$

and so $V_{N}$ grows linearly fast in $N$.

Next, since we can take $K=3$, the denominators $m$ of the nonzero resonant
points can only be $2,3,4,5$ or $6$. An easy check shows that for $m=3,5,6$ we
have $q_{n}(m)\geq\rho$, and that for $m=2,4$ we have $q_{n}(m)=b_{n}$.
Therefore, for $m=3,5,6$ we have $M_{N}(m)\geq\rho N$, and so we can disregard
all the nonzero resonant points except for $\pi/2,\pi$ and $3\pi/2$. For the
latter points we have

(10.6) $\phi_{n}\left(\frac{\pi}{2}\right)=b_{n}-i(1-b_{n}),$ (10.7)
$\phi_{n}(\pi)=2b_{n}-1,\quad\phi_{n}\left(\frac{3\pi}{2}\right)=b_{n}+i(1-b_{n}).$

Hence, denoting $\eta_{n}=b_{n}(1-b_{n}),$ we have

$\left|\phi_{n}\left(\frac{\pi}{2}\right)\right|^{2}=\left|\phi_{n}\left(\frac{3\pi}{2}\right)\right|^{2}=1-2\eta_{n},\quad\big{|}\phi_{n}(\pi)\big{|}^{2}=1-4\eta_{n}.$

Since we suppose that $\eta_{n}\leq b_{n}<\frac{1}{8}$ it follows that
$1-4\eta_{n}\geq\frac{1}{2}$. Then for the above three resonant points we can
take $N_{0}=0$. Now Proposition 7.1 and a simple calculation show that for any
$r$ we get the Edgeworth expansions of order $r$ if and only if

$\prod_{n=1}^{N}(1-2\eta_{n})=o\left(N^{1-r}\right).$

Let us focus for the moment on the case when $b_{n}=\gamma/n$ for $n$ large
enough where $\gamma>0$ is a constant. Rewriting (10.6), (10.7) as

(10.8)
$\frac{\phi_{n}\left(\frac{\pi}{2}\right)}{-i}=(1-b_{n})+ib_{n},\quad\frac{\phi_{n}\left(\frac{3\pi}{2}\right)}{i}=(1-b_{n})-ib_{n},\quad$
$-\phi_{n}(\pi)=1-2b_{n}$

and, using that the condition $b_{n}<\frac{1}{8}$ implies that that
$\phi_{n}(t)\neq 0$ for all $n\in\mathbb{N}$ and all
$t\in\left\\{\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\\}$, we conclude similarly
to Example 10.3 that there are non-zero complex numbers
$\kappa_{1},\kappa_{3}$ and a non-zero real number $\kappa_{2}$ such that

$\Phi_{N}\left(\frac{\pi}{2}\right)=\frac{(-i)^{N}\kappa_{1}}{N^{\gamma}}e^{i\gamma\ln
N}\left(1+O\left(\frac{1}{N}\right)\right),$
$\Phi_{N}\left(\frac{3\pi}{2}\right)=\frac{i^{N}\kappa_{3}}{N^{\gamma}}e^{-i\gamma\ln
N}\left(1+O\left(\frac{1}{N}\right)\right),$
$\Phi_{N}(\pi)=(-1)^{N}\frac{\kappa_{2}e^{2\gamma
w_{N}}}{N^{2\gamma}}\left(1+O\left(\frac{1}{N}\right)\right).$

It follows that $S_{N}$ admits Edgeworth expansion of order $r$ iff
$\gamma>\frac{r-1}{2}.$ In fact if $\frac{r-1}{2}<\gamma\leq\frac{r}{2}$ then
Corollary 7.4 shows that

$\mathbb{P}(S_{N}=k)=\frac{e^{-k_{N}^{2}/2}}{\sqrt{2\pi}}\Bigg{[}{\mathcal{E}}_{r}(k_{N})+\frac{\kappa_{1}e^{i\gamma\ln
N}}{N^{\gamma}{\sigma}_{N}}+\frac{\kappa_{3}e^{-i\gamma\ln
N}}{N^{\gamma}{\sigma}_{N}}+O\left(N^{-\eta}\right)\Bigg{]}$

where ${\mathcal{E}}_{r}$ is the Edgeworth polynomial of order $r$ and
$\eta\\!\\!=\\!\\!\min\left(2\gamma,\frac{r}{2}\right)+\frac{1}{2}.$

To give a specific example, let us suppose that $\frac{1}{2}\leq\gamma<1$ and
that $E(X_{n})=0$ which means that

(10.9) $a_{n}=\frac{3(1-b_{n})}{4},\quad c_{n}=\frac{1-b_{n}}{4}.$

Then

(10.10) $V_{N}=3N-3\gamma\ln N+O(1),\quad E(S_{N}^{3})=6N-6\gamma\ln N+O(1),$

so Proposition 9.1 gives

$\sqrt{2\pi}\mathbb{P}(S_{N}=k)=$
$e^{-k^{2}/6N}\left[\frac{1}{\sqrt{3N}}\left(1+\frac{\kappa_{1}i^{k-N}e^{i\gamma\ln
N}+\kappa_{3}i^{N-k}e^{-i\gamma\ln
N}}{N^{\gamma}}\right)-\frac{k^{3}}{81\sqrt{3N^{5}}}\right]$
$+O\left(N^{-3/2}\right)$

Next, let us provide the second order trigonometric expansions under the sole
assumption that $1-4\eta_{n}\geq\frac{1}{2}$ and $a_{n},c_{n}\geq\rho$. As we
have mentioned, we only need to consider the nonzero resonant points
$\pi/2,\pi,3\pi/2$ and for these points we have $N_{0}=0$. Therefore, the term
involving the derivative in the right hand side of (9.1) vanishes. Now, a
direct calculation shows that

$C_{1,N,\pi}=\sum_{n=1}^{N}\frac{{\mathbb{E}}(e^{i\pi
X_{n}}\bar{X}_{n})}{{\mathbb{E}}(e^{i\pi
X_{n}})}=2\sum_{n=1}^{N}\frac{(a_{n}-3c_{n})b_{n}}{2b_{n}-1}$

and

$C_{1,N,\pi/2}=\sum_{n=1}^{N}\frac{(a_{n}-3c_{n})(1+i)b_{n}}{b_{n}-i(1-b_{n})},\quad
C_{1,N,3\pi/2}=\sum_{n=1}^{N}\frac{(a_{n}-3c_{n})(1-i)b_{n}}{b_{n}+i(1-b_{n})}.$

Note that $3c_{n}-a_{n}={\mathbb{E}}(X_{n})$. Set

$\Gamma_{1,N}=\prod_{n=1}^{N}(b_{n}-i(1-b_{n})),\quad\Gamma_{2,N}=\prod_{n=1}^{N}(2b_{n}-1),\quad\Gamma_{3,N}=\prod_{n=1}^{N}(b_{n}+i(1-b_{n})).$

Then $\Gamma_{s,N}={\mathbb{E}}(e^{\frac{s\pi i}{2}S_{N}})$. We also set

$\Theta_{s,N}=C_{1,N,s\pi/2}\Gamma_{s,N},\,s=1,2,3$

and

$\Gamma_{N}(k)=\sum_{j=1}^{3}e^{-j\pi
ik/2}\Gamma_{j,N},\quad\Theta_{N}(k)=\sum_{j=1}^{3}e^{-j\pi
ik/2}\Theta_{j,N}.$

Then by Proposition 9.1 and Remark 9.2, uniformly in $k$ we have

(10.11)
$\sqrt{2\pi}P(S_{N}=k)={\sigma}_{N}^{-1}\left(1+\Gamma_{N}(k)\right)e^{-k_{N}^{2}/2}$
$-{\sigma}_{N}^{-2}\left(k_{N}^{3}T_{N}\big{(}1+\Gamma_{N}(k)\big{)}+ik_{N}\Theta_{N}(k)\right)e^{-k_{N}^{2}/2}+o({\sigma}_{N}^{-2})$

where
$T_{N}=\frac{\displaystyle\sum_{n=1}^{N}{\mathbb{E}}(\bar{X}_{n}^{3})}{6V_{N}}$,
$\bar{X}_{n}=X_{n}-{\mathbb{E}}(X_{n})$.

Let us now consider a more specific situation. Namely we suppose that
$b_{n}=\frac{\gamma}{n^{3/2}}$ for large $n$ and that $E(X_{n})=0.$ Then
(10.9) shows that $C_{1,N,s\pi/2}=0.$ Next (10.8) gives

$\frac{\Phi_{N}\left(\frac{\pi}{2}\right)}{(-i)^{N}}=\prod_{n=1}^{N}\left[(1-b_{n})+ib_{n}\right]=\frac{{\bar{\kappa}}_{1}}{\displaystyle\prod_{n=N+1}^{\infty}\left[(1-b_{n})+ib_{n}\right]}$
$={\bar{\kappa}}_{1}\left(1+\frac{2\gamma(1-i)}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right)$

where
$\displaystyle{\bar{\kappa}}_{1}=\prod_{n=1}^{\infty}\left[(1-b_{n})+ib_{n}\right].$
Likewise

$\frac{\Phi_{N}\left(\frac{3\pi}{2}\right)}{i^{N}}={\bar{\kappa}}_{3}\left(1+\frac{2\gamma(1+i)}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right)$

and

$\frac{\Phi_{N}\left(\pi\right)}{(-1)^{N}}={\bar{\kappa}}_{2}\left(1+\frac{4\gamma}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right).$

Taking into account (10.10) we can reduce (10.11) to the following expansion

$\sqrt{2\pi}{\mathbb{P}}(S_{N}=k)=e^{-k^{2}/6N}\left[\frac{1}{\sqrt{3N}}\left(1+\sum_{s=1}^{3}{\bar{\kappa}}_{s}i^{s(k-N)}\right)\right.$
$\left.+\frac{1}{N}\left(-\frac{\tilde{k}_{N}^{3}}{3}+\sum_{s=1}^{3}{\bar{\kappa}}_{s}i^{s(k-N)}\left(\frac{2\gamma(1-i^{-s})}{\sqrt{3}}-\frac{\tilde{k}_{N}^{3}}{3}\right)\right)\right]+O\left(\frac{1}{N^{3/2}}\right)$

where $\tilde{k}_{N}=k/\sqrt{3N}$.

######  Example 10.5.

Let $X^{\prime}$ take value $\pm 1$ with probability $\frac{1}{2}$,
$X^{\prime\prime}$ take values $0$ and $1$ with probability $\frac{1}{2}$, and
$X^{\delta}$, ${\delta}\in[0,1]$ be the mixture of $X^{\prime}$ and
$X^{\prime\prime}$ with weights $\delta$ and $1-\delta.$ Thus $X^{\delta}$
take value $-1$ with probability $\frac{\delta}{2},$ the value $0$ with
probability $\frac{1-\delta}{2}$ and value $1$ with probability $\frac{1}{2}$.
Therefore, ${\mathbb{E}}(e^{\pi iX^{\delta}})=-{\delta}$. We suppose that
$X_{2m}$ and $X_{2m-1}$ have the same law which we call $Y_{m}.$ The
distribution of $Y_{m}$ is defined as follows. Set $k_{j}=3^{3^{j}}$, and let
$Y_{k_{j}}$ have the same distribution as $X^{{\delta}_{j}}$ where
${\delta}_{j}=\frac{1}{\sqrt{k_{j+1}}}$. When $m\not\in\\{k_{j}\\}$ we let
$Y_{m}$ have the distribution of $X^{\prime}$. It is clear that $V_{N}$ grows
linearly fast in $N$. Note also that ${\mathbb{E}}(e^{\pi
iY_{m}})=-{\delta}_{j}$ when $m=k_{j}$ for some $j$, and otherwise
${\mathbb{E}}(e^{\pi iY_{m}})=-1$. Now, take $N\in{\mathbb{N}}$ such that
$N>2k_{2}$, and let $J_{N}$ be so that $2k_{J_{N}}\leq N<2k_{J_{N}+1}$. Then

$|\Phi_{N}(\pi)|\leq\prod_{j=1}^{J_{N}}(k_{j+1})^{-1}.$

Since $k_{J_{N}}\leq\frac{N}{2}<k_{J_{N}+1}$ and $k_{j}=(k_{j+1})^{1/3}$ we
have $k_{J_{N}+1}^{-1}\leq 2N^{-1}$ and $k_{J_{N}+1-m}\leq
2^{3^{-m}}N^{-3^{-m}}$ for any $0<m\leq J_{N}$. Denote
$\displaystyle\alpha_{N}=\sum_{j=1}^{J_{N}-1}3^{-j}.$ Since $\alpha_{N}>1/3$
we get that

$|\Phi_{N}(\pi)|\leq 2^{3/2}N^{-\alpha_{N}}=o(N^{-1-1/3}).$

Similarly, for each $j_{1},j_{2}\leq N$,

(10.12) $|\Phi_{N:j_{1}}(\pi)|\leq 2^{3/2}N^{-1/2-\alpha_{N}}=o(N^{-1/2-1/3})$

and

(10.13) $|\Phi_{N:j_{1},j_{2}}(\pi)|\leq 2^{3/2}N^{-\alpha_{N}}=o(N^{-1/3}).$

Indeed, the largest possible values are obtained for $j_{1}=2k_{J_{N}}$ (or
$j_{1}=2k_{J_{N+1}}-1$ if it is smaller than $N+1$) and $j_{2}=2k_{J_{N}}-1$
(or $j_{2}=2k_{J_{N}}$). Using the same estimates as in the proof of of
Theorem 1.8 we conclude from (10.12) that
$\displaystyle\Phi_{N}^{\prime}(\pi)=o\left(1/\sqrt{N}\right)$ and we conclude
from (10.13) that $\displaystyle\Phi_{N}^{\prime\prime}(\pi)=o(1).$ It follows
from Lemma 7.6 and Proposition 7.1 that $S_{N}$ satisfies an Edgeworth
expansion of order 3. The same conclusion holds if we remove a finite number
of terms from the beginning of the sequence $\\{X_{n}\\}$ because the
smallness of $\Phi_{N}(\pi)$ comes from the terms $X_{2k_{j}-1}$ and
$X_{2k_{j}}$ for arbitrary large $j$’s.

On the other hand

$\left|\Phi_{2k_{j};2k_{j},2k_{j}-1,2k_{j-1},2k_{j-1}-1}(\pi)\right|=\prod_{s=2}^{j-1}\left(3^{-3^{s}}\right)$
$=3^{-(3^{j}-9)/2}=\frac{3^{9/2}}{\sqrt{k_{j}}}\gg\frac{1}{k_{j}}=3^{-3^{j}}.$

It follows that $S_{2k_{j};2k_{j},2k_{j}-1,2k_{j-1},2k_{j-1}-1}$ does not obey
the Edgeworth expansion of order 3. Accordingly, stable Edgeworth expansions
need not be superstable if $r=3.$ A similar argument allows to construct
examples showing that those notions are different for all $r>2.$

##  11\. Extension for uniformly bounded integer-valued triangular arrays

In this section we will describe our results for arrays of independent random
variables. We refer to [13], [26, 27] and [11], [35], [29] and [10] for
results for triangular arrays of inhomogeneous Markov chains. Example where
Markov arrays appear naturally include the theory of large deviations for
inhomogeneous systems (see [34, 31, 14] and references wherein), random walks
in random scenery [4, 16], and statistical mechanics [19].

Let $X_{n}^{(N)},\,1\leq n\leq L_{N}$ be a triangular array such that for each
fixed $N$, the random variables $X_{n}^{(N)}$ are independent and integer
valued. Moreover, we assume that

$K:=\sup_{N}\sup_{n}\|X_{n}\|_{L^{\infty}}<\infty.$

For each $N$ we set $\displaystyle S_{N}=\sum_{n=1}^{L_{N}}X_{n}^{(N)}$. Let
$V_{N}=\text{Var}(S_{N})$. We assume that $V_{N}\to\infty$, so that, by
Lindenberg–Feller Theorem, the sequence
$(S_{N}-{\mathbb{E}}(S_{N}))/{\sigma}_{N}$ obeys the CLT, where
${\sigma}_{N}=\sqrt{V_{N}}$.

We say that the array $X_{n}^{(N)}$ obeys the SLLT if for any $k$ the LLT
holds true for any uniformly square integrable array $Y_{n}^{(N)},\,1\leq
n\leq L_{N}$, so that $Y_{n}^{(N)}=X_{n}^{(N)}$ for all but $k$ indexes $n$.
Set

$M_{N}:=\min_{2\leq h\leq 2K}\sum_{n=1}^{L_{N}}P(X_{n}\neq
m_{n}^{(N)}(h)\text{ mod }h)\geq R\ln V_{N}$

where $m_{n}^{(N)}(h)$ is the most likely value of $X_{n}^{(N)}$ modulo $h$.
Observe now that the proofs of Proposition 3.1 and Lemmas 3.3, 3.4, 4.2 and
4.4 proceed exactly the same for arrays. Therefore, all the arguments in the
proof of Theorem 2.1 proceed the same for arrays instead of a fixed sequence
$X_{n}$. That is, we have

######  Theorem 11.1.

There $\exists J=J(K)<\infty$ and polynomials $P_{a,b,N}$ with degrees
depending only on $a$ and $b$, whose coefficients are uniformly bounded in $N$
such that, for any $r\geq 1$ uniformly in $k\in{\mathbb{Z}}$ we have

${\mathbb{P}}(S_{N}=k)-\sum_{a=0}^{J-1}\sum_{b=1}^{r}\frac{P_{a,b,N}((k-a_{N})/\sigma_{N})}{\sigma_{N}^{b}}\mathfrak{g}((k-a_{N})/\sigma_{N})e^{2\pi
iak/J}=o(\sigma_{N}^{-r})$

where $a_{N}={\mathbb{E}}(S_{N})$ and
$\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}}e^{-u^{2}/2}.$

Moreover, $P_{0,1,N}\equiv 1$ and given $K,r$, there exists $R=R(K,r)$ such
that if $M_{N}\geq R\ln V_{N}$ then we can choose $P_{a,b,N}=0$ for $a\neq 0.$

All the formulas for the coefficients of the polynomials $P_{a,b,N}$ remain
the same in the arrays setup. In particular, we get that, uniformly in $k$ we
have

(11.1)
${\mathbb{P}}(S_{N}=k)=\left(1+\sum_{t\in{\mathcal{R}}}e^{-itk}\Phi_{N}(t)\right)e^{-k_{N}^{2}/2}{\sigma}_{N}^{-1}+o({\sigma}_{N}^{-1})$

where $\Phi_{N}(t)={\mathbb{E}}(e^{itS_{N}})$.

Next, our version for Proposition 1.2 for arrays is as follows.

######  Proposition 11.2.

Suppose $S_{N}$ obeys LLT. Then for each integer $h\geq 2$, at least one of
the following conditions occur: either (a)
$\displaystyle\lim_{N\to\infty}\sum_{n=1}^{L_{N}}{\mathbb{P}}(X_{n}\neq
m_{n}^{(N)}(h)\text{ mod }h)=\infty$. or (b) there exists a subsequence
$N_{k}$, numbers $s\in{\mathbb{N}}$ and ${\varepsilon}_{0}>0$ and indexes
$1\leq j_{1}^{k},...,j_{s_{k}}^{k}\leq L_{N_{k}}$, $s_{k}\leq s$ so that the
distribution of $\displaystyle\sum_{u=1}^{s_{k}}X^{(N_{k})}_{j_{u}}$ converges
to uniform $\text{mod }h$, and the distance between the distribution of
$\displaystyle S_{N_{k}}-\sum_{q=1}^{s_{k}}X^{(N_{k})}_{j_{q}}$ and the
uniform distribution $\text{mod }h$ is at least ${\varepsilon}_{0}$.

###### Proof.

First, by (11.1) and Lemma 5.1 if the LLT holds then for any nonzero resonant
point $t$ we have $\displaystyle\lim_{N\to\infty}|\Phi_{N}(t)|=0$. Now, if (a)
does not hold true then there is a subsequence $N_{k}$ so that
$\displaystyle\sum_{n=1}^{L_{N_{k}}}q(X_{n}^{(N_{k})},h)\leq C$, where $C$ is
some constant. Set $\displaystyle
q_{n}^{(N_{k})}(h)=q\left(X_{n}^{(N_{k})},h\right)$. Then there are at most
$8hC$ $n$’s between $1$ and $L_{N_{k}}$ so that
$q_{n}^{(N_{k})}(h)>\frac{1}{8h}$. Let us denote these $n$’s by
$n_{1,k},...,n_{s_{k},k}$, $s_{k}\leq 8hC$. Next, for any $n$ and a nonzero
resonant point $t=2\pi l/h$ we have

(11.2) $|\phi_{n}^{(N_{k})}(t)|\geq 1-2hq_{n}^{(N_{k})}(h)\geq e^{-2\gamma
hq_{n}^{(N_{k})}}$

where $\phi_{n}^{(N_{k})}$ is the characteristic function of $X_{n}^{(N_{k})}$
and $\gamma$ is such that for $\theta\in[0,1/4]$ we have $1-\theta\geq
e^{-\gamma\theta}.$ We thus get that

(11.3)
$\prod_{n\not\in\\{n_{u,k}\\}}|\phi_{n}^{(N_{k})}(t)|\geq\prod_{n\not\in\\{n_{u,k}\\}}(1-2hq_{n}^{(N_{k})}(h))\geq
C_{0}$

where $C_{0}>0$ is some constant. Therefore,

$|\Phi_{N_{k}}(t)|\geq\prod_{u=1}^{s_{k}}|\phi_{n_{u,k}}^{(N_{k})}(t)|\cdot
C_{0}$

and so we must have

(11.4) $\lim_{k\to\infty}\prod_{u=1}^{s_{k}}|\phi_{n_{u,k}}^{(N_{k})}(t)|=0.$

Now (b) follows from (11.3), (11.4) and Lemma 6.2. ∎

Using (11.1) we can now prove a version of Theorem 1.1 for arrays.

######  Theorem 11.3.

The SLLT holds iff for each integer $h>1$,

(11.5) $\lim_{N\to\infty}\sum_{n=1}^{L_{N}}{\mathbb{P}}(X_{n}^{(N)}\neq
m_{n}\text{ mod }h)=\infty$

where $m_{n}=m_{n}^{(N)}(h)$ is the most likely residue of $X_{n}^{(N)}$
modulo $h$.

###### Proof.

First, the arguments in the proof of (3.8) show that there are constants
$c_{0},C>0$ so that for any nonzero resonant point $t=2\pi l/h$ we have

(11.6) $|\Phi_{N}(t)|\leq Ce^{-c_{0}M_{N}(h)},\quad\text{where}\quad
M_{N}(h):=\sum_{n=1}^{L_{N}}q(X_{n}^{(N)},h).$

Let us assume that (11.5) holds for all integers $h>1$. Consider
$s_{N}$–tuples $1\leq j_{1}^{N},...,j_{s_{N}}^{N}\leq L_{N}$, where
$s_{N}\leq\bar{s}$ is bounded in $N$. Then by applying (11.6) with
$\displaystyle\tilde{S}_{N}=S_{N}-\sum_{l=1}^{s_{N}}X_{j_{l}^{N}}^{(N)}$ we
have

(11.7) $\lim_{N\to\infty}|{\mathbb{E}}(e^{it\tilde{S}_{N}})|=0.$

Now, arguing as in the proof of Theorem 1.8(1), given a uniformly square
integrable array $Y_{n}^{(N)}$ as in the definition of the SLLT, we still have
(11.1), even though the new array is not necessarily uniformly bounded.
Applying (11.7) we see that for any nonzero resonant point $t$ we have

$\lim_{N\to\infty}\left|{\mathbb{E}}\left(\exp\left[it\sum_{n=1}^{L_{N}}Y_{n}^{(N)}\right]\right)\right|=0$

and so $\displaystyle S_{N}Y:=\sum_{n=1}^{L_{N}}Y_{n}^{(N)}$ satisfies the
LLT.

Now let us assume that $M_{N}(h)\not\to\infty$ for some $2\leq h\leq 2K$ (it
not difficult to see that (11.5) holds for any $h>2K$).

In other words after taking a subsequence we have that $M_{N_{k}}(h)\leq L$
for some $L<\infty.$ The proof of Proposition 11.2 shows that there $s<\infty$
such that after possibly removing terms $n_{1,k},n_{2,k},\dots,n_{s_{k},k}$
with $s_{k}\leq s$ we can obtain that $q_{n}^{(N_{k})}(h)\leq\frac{1}{8h},$
$n\not\in\\{n_{j,k}\\}$. In this case (11.2) shows that for each $\ell$

$|\Phi_{N_{k};n_{1,k},\dots,n_{s_{k},k}}(2\pi\ell/h)|\geq e^{-2\gamma L}.$

By Proposition 11.2, $S_{N_{k};n_{1,k},\dots,n_{s_{k},k}}$ does not satisfy
the LLT. ∎

Next, all the other arguments in our paper proceed similarly for arrays since
they essentially rely only on the specific structure of the polynomials from
Theorem 2.1. For the sake of completeness, let us formulate the main
(remaining) results here.

######  Theorem 11.4.

The following conditions are equivalent:

(a) $S_{N}$ satisfies LLT;

(b) For each $\xi\in{\mathbb{R}}\setminus{\mathbb{Z}}$,
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left(e^{2\pi i\xi
S_{N}}\right)=0$;

(c) For each non-zero resonant point $\xi$,
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left(e^{2\pi i\xi
S_{N}}\right)=0$;

(d) For each integer $h$ the distribution of $S_{N}$ mod $h$ converges to
uniform.

######  Theorem 11.5.

For each $r$ there is $R=R(r,K)$ such that the Edgeworth expansion of order
$r$ holds true if $M_{N}\geq R\ln V_{N}$. In particular, $S_{N}$ obeys
Edgeworth expansions of all orders if

$\lim_{N\to\infty}\frac{M_{N}}{\ln V_{N}}=\infty.$

######  Theorem 11.6.

For any $r\geq 1$, the Edgeworth expansion of order $r$ holds if and only if
for any nonzero resonant point $t$ and $0\leq\ell<r$ we have

$\bar{\Phi}_{N}^{(\ell)}(t)=o\left({\sigma}_{N}^{\ell+1-r}\right)$

where $\bar{\Phi}_{N}(x)={\mathbb{E}}[e^{ix(S_{N}-{\mathbb{E}}(S_{N}))}]$.

######  Theorem 11.7.

Suppose $S_{N}$ obeys the SLLT. Then the following are equivalent:

(a) Edgeworth expansion of order 2 holds;

(b) $|\Phi_{N}(t)|=o(\sigma_{N}^{-1})$ for each nonzero resonant point $t$;

(c) For each $h\leq 2K$ the distribution of $S_{N}$ mod $h$ is
$o(\sigma_{N}^{-1})$ close to uniform.

Next, we say that an array $\\{X_{n}^{(N)}\\}$ admits an Edgeworth expansion
of order $r$ in a superstable way (denoted by $\\{X_{n}^{(N)}\\}\in EeSs(r)$)
if for each ${\bar{s}}$ and each sequence
$j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$ with $s_{N}\leq{\bar{s}}$ and
$j_{i}^{N}\leq L_{N}$ there are polynomials $P_{b,N}$ whose coefficients are
$O(1)$ in $N$ and their degrees do not depend on $N$ so that uniformly in
$k\in{\mathbb{Z}}$ we have that

(11.8)
${\mathbb{P}}(S_{N;j_{1}^{N},j_{2}^{N},\dots,j_{{s_{N}}^{N}}}=k)=\sum_{b=1}^{r}\frac{P_{b,N}(k_{N})}{\sigma_{N}^{b}}\mathfrak{g}(k_{N})+o(\sigma_{N}^{-r})$

and the estimates in $O(1)$ and $o(\sigma_{N}^{-r})$ are uniform in the choice
of the tuples $j_{1}^{N},\dots,j_{s_{N}}^{N}.$

Let $\Phi_{N;j_{1},j_{2},\dots,j_{s}}(t)$ be the characteristic function of
$S_{N;j_{1},j_{2},\dots,j_{s}}.$

######  Theorem 11.8.

(1) $S_{N}\in EeSs(1)$ (that is, $S_{N}$ satisfies the LLT in a superstable
way) if and if it satisfies the SLLT.

(2) For arbitrary $r\geq 1$ the following conditions are equivalent:

(a) $\\{X_{n}^{(N)}\\}\in EeSs(r)$;

(b) For each $j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$ and each nonzero
resonant point $t$ we have
$\Phi_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}(t)=o(\sigma_{N}^{1-r});$

(c) For each $j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}$, and each $h\leq 2K$
the distribution of $S_{N;j_{1}^{N},j_{2}^{N},\dots,j_{s_{N}}^{N}}$ mod $h$ is
$o(\sigma_{N}^{1-r})$ close to uniform.

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