# On the rate of convergence of the martingale central limit theorem in
Wasserstein distances

Xiaoqin Guo The work of XG is supported by Simons Foundation through
Collaboration Grant for Mathematicians #852943.  Department of Mathematical
Sciences, University of Cincinnati , 2815 Commons Way, Cincinnati, OH 45221,
USA

###### Abstract

For martingales with a wide range of integrability, we will quantify the rate
of convergence of the central limit theorem via Wasserstein distances of order
$r$, $1\leq r\leq 3$. Our bounds are in terms of Lyapunov’s coefficients and
the $\mathscr{L}^{r/2}$ fluctuation of the total conditional variances. We
will show that our Wasserstein-1 bound is optimal up to a multiplicative
constant.

###### Contents

  1. 1 Introduction
    1. 1.1 Earlier results in the literature
    2. 1.2 Motivation and our contributions
    3. 1.3 Structure of the paper
    4. 1.4 Preliminaries: N-functions and an Orlicz-norm for sequences
  2. 2 Main results
  3. 3 Martingales with $V_{n}^{2}=1$
    1. 3.1  Wasserstein distance bounds via Lindeberg’s argument
    2. 3.2 The rates of some special cases
  4. 4 Proofs of the main $\mathcal{W}_{r}$ bounds
    1. 4.1 A modified martingale, and Proof of Theorem 8 ($\mathcal{W}_{3}$ bounds)
    2. 4.2 Proof of Theorem 5 ($\mathcal{W}_{1}$ bounds)
    3. 4.3 Proof of Theorem 7 ($\mathcal{W}_{2}$ bounds)
  5. 5 Optimality of the $\mathcal{W}_{1}$ rates: Proof of Proposition 10
  6. 6 Some open questions
  7. A Appendix
    1. A.1 Comparison between Orlicz norms
    2. A.2 Regularity of Gaussian smoothing:Proof of Lemma 12
    3. A.3 Moment bound of the maximum
    4. A.4 Proof of Lemma 20
    5. A.5 A Burkholder inequality

##  1 Introduction

In this paper, we consider the rate of convergence of the martingale central
limit theorem (CLT) under Wasserstein distances. Let
$Y_{\cdot}=\\{Y_{m}\\}_{m=1}^{n}$ be a square integrable martingale difference
sequence (mds) with respect to $\sigma$-fields
$\\{\mathcal{F}_{m}\\}_{m=0}^{n}$. Here $\mathcal{F}_{0}$ denotes the trivial
$\sigma$-field. For $1\leq m\leq n$, let
$s_{m}^{2}=\sum_{i=1}^{m}E[Y_{i}^{2}]$ and

$\displaystyle\sigma_{m}^{2}(Y)$
$\displaystyle=E[Y_{m}^{2}|\mathcal{F}_{m-1}],$ $\displaystyle V_{n}^{2}$
$\displaystyle=\frac{1}{s_{n}^{2}}\sum_{i=1}^{n}\sigma_{i}^{2}(Y),$
$\displaystyle X_{m}$ $\displaystyle=\frac{1}{s_{n}}\sum_{i=1}^{m}Y_{i}.$ (1)

For a mds $\\{Y_{i}\\}_{i=1}^{\infty}$, when $\lim_{n\to\infty}V_{n}^{2}=1$ in
probability and Lindeberg’s condition is satisfied, it is well-known that

$X_{n}\Rightarrow\mathcal{N}\quad\text{ as }n\to\infty,$

where $\mathcal{N}\sim\mathcal{N}(0,1)$ denotes a standard normal random
variable.

To quantify such a convergence in distribution, one of the most important
metrics is the Wasserstein distance which has deep roots in the optimal
transport theory [42]. Recall that, for two probability measures $\mu,\nu$ on
$\mathbb{R}$, their Wasserstein distance (also called minimal distance)
$\mathcal{W}_{r}$ of order $r$, $r\geq 1$, is defined by

$\mathcal{W}_{r}(\mu,\nu)=\inf\\{\lVert U-V\rVert_{r}:U\sim\mu,V\sim\nu\\},$

where $\lVert U\rVert_{r}=E[|U|^{r}]^{1/r}$. With abuse of notations, we may
use random variables as synonyms of their distributions. E.g., for
$U\sim\mu,V\sim\nu$, we may write $\mathcal{W}_{r}(\mu,\nu)$ as
$\mathcal{W}_{r}(U,V)$.

When $r=1$, recall that $\mathcal{W}_{1}$ admits the following alternative
representations:

$\mathcal{W}_{1}(U,V)=\sup_{h\in\rm{Lip}_{1}}\left\\{E[h(U)]-E[h(V)]\right\\}=\int_{\mathbb{R}}\Bigr{\lvert}F_{U}(x)-F_{V}(x)\Bigl{\rvert}\mathrm{d}x$

where $\rm{Lip}_{1}$ denotes the set of all 1-Lipschitz functions, and
$F_{U}(x):=P(U\leq x)$.

Throughout the paper, we use $c,C$ to denote positive constants which may
change from line to line. Unless otherwise stated, $C_{p},c_{p}$ denote
constants depending only on the parameter $p$. We write $A\lesssim B$ if
$A\leq CB$, and $A\asymp B$ if $A\lesssim B$ and $A\gtrsim B$. We also use
notations $A\lesssim_{p}B$, $A\asymp_{p}B$ to indicate that the multiplicative
constant depends on the parameter $p$.

###  1.1 Earlier results in the literature

There is an immense literature on the convergence rate of the CLT for
independent random variables. Such results are often phrased in terms of
Lyapunov’s coefficient (i.e., the term in the $\mathscr{L}^{p}$ Lyapunov
condition). To be specific, for $p\geq 1$, set111Note that the meaning of our
$L_{p}$ notation is slightly different from that of [2, 27, 25, 38, 4] in the
literature whose $L_{p}$ means the $L_{p}^{p}$ in our paper.

$L_{p}=L_{p}(Y):=\frac{1}{s_{n}}\left(\sum_{i=1}^{n}E[|Y_{i}|^{p}]\right)^{1/p}.$
(2)

Note that typically $L_{p}$ is of size $O(n^{1/p-1/2})$.

When $\\{Y_{i}\\}_{i=1}^{n}$ is a centered independent sequence, a nonuniform
estimate of Bikjalis [2] implies $\mathcal{W}_{1}(X_{n},\mathcal{N})\leq
cL_{p}^{p}$, $p\in(2,3]$, extending the $\mathcal{W}_{1}$ bounds of Esseen
[16], Zolotarev [45], and Ibragimov [29]. For $r>2$, Sakhanenko [40] proved
that $\mathcal{W}_{r}(X_{n},\mathcal{N})\leq cL_{r}$ which is optimal for
independent $\mathscr{L}^{r}$-integrable variables. For $1\leq r\leq 2$, it is
established by Rio [37, 38] that

$\mathcal{W}_{r}(X_{n},\mathcal{N})\leq c_{r}L_{r+2}^{(r+2)/r}.$ (3)

In particular, when $\\{Y_{i}\\}_{i=1}^{n}$ have roughly the same
$\mathscr{L}^{r+2}$ moments, the $\mathcal{W}_{r}$ bound in (3) achieves the
optimal rate $O(n^{-1/2})$. Bobkov [4] confirmed Rio’s conjecture that (3)
holds for all $r\geq 1$. Cf. [38, 4] and references therein. For further
developments, see e.g. [44, 20, 8, 22, 14, 6, 33] for work in the multivariate
setting and e.g. [21, 33] for results on random variables with local-
dependence.

It is natural to to ask whether the martingale CLT can be quantified by
similar Wasserstein metric bounds in terms of $L_{p}$. However, despite its
theoretical importance, there are very few such results for (general)
martingales compared to the independent case, let alone questions on optimal
rates.

When $\\{Y_{i}\\}_{i=1}^{n}$ is a mds, the non-uniform bound of the
distribution functions by Joos [31] (which generalizes Haeusler and Joos [26])
implies that

$\mathcal{W}_{1}(X_{n},\mathcal{N})\leq c_{p,q}\left(L_{p}^{p/(1+p)}+\lVert
V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}\right)\quad\text{ for }2<p<\infty,q\geq 1.$
(4)

In the case $M=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}<\infty$, the
nonuniform bound of Joos [30] implies

$\mathcal{W}_{1}(X_{n},\mathcal{N})\leq
c_{q}\left(\frac{M}{s_{n}}\log\big{(}e+\frac{s_{n}^{2}}{M^{2}}\big{)}+\lVert
V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}\right)\,\text{ for }q\geq 1.$ (5)

Still for the $\mathscr{L}^{\infty}$ case, Dung, Son, Tien [12] improved the
last term in (5) to be $C_{q}\lVert V^{2}-1\rVert_{q}^{1/2}$ for any $q>1/2$.
Under the condition

$P(V_{n}^{2}=1)=1,$ (6)

Röllin’s [39, (2.1)] result inferred that (See also [18, Lemma 2.1])

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{3}.$ (7)

Fan and Ma [18] dropped the condition (6) and obtained

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{q}L_{3}+\lVert
V_{n}^{2}-1\rVert_{q}^{1/2}+\frac{1}{s_{n}}\max_{i=1}^{n}\lVert
Y_{i}\rVert_{2q},\quad\forall q\geq 1.$ (8)

Recently, assuming (6), Fan, Su [19, Corollary 2.5] implicitly extended (7) to
be

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{p}\quad\text{ for }2<p\leq 3.$
(9)

Dedecker, Merlevède, and Rio [11] proved $\mathcal{W}_{r}$ bounds,
$r\in(0,3]$, that involve the quantities
$\sum_{i=\ell}^{n}E[Y_{i}^{2}|\mathcal{F}_{\ell-1}]-E[Y_{i}^{2}]$ (instead of
$V_{n}^{2}-1$ in our bounds), $2\leq\ell\leq n$, which can be suitably bounded
in many situations.

Readers may refer to [10, 9, 39, 19, 11] for Wasserstein bounds for the
martingale CLT under other special conditions (e.g. the sequence being
stationary, $\sigma_{i}$’s being close to be deterministic, $\sigma_{i}$’s are
uniformly bounded from below, or certain variants of the
$\mathscr{L}^{\infty}$ case, etc.).

######  Remark 1.

Unlike the Wasserstein metric, the Kolmogorov distance bounds for the
martingale CLT has been thoroughly investigated since the 1970s. One of the
earliest results is due to Heyde, Brown [28] which states that, for $2<p\leq
4$,

$d_{K}(X_{n},\mathcal{N}):=\sup_{x\in\mathbb{R}}\lvert
F_{X_{n}}(x)-F_{\mathcal{N}}(x)\rvert\lesssim_{p}L_{p}^{p/(p+1)}+\lVert
V_{n}^{2}-1\rVert_{p/2}^{p/(2p+2)}.$ (10)

When the mds is $\mathscr{L}^{\infty}$, i.e., $\max_{i=1}^{n}\lVert
Y_{i}\rVert_{\infty}=M<\infty$, Bolthausen [5] showed that

$d_{K}(X_{n},\mathcal{N})\lesssim_{M}\frac{n\log
n}{s_{n}^{3}}+\min\bigg{\\{}\lVert V_{n}^{2}-1\rVert_{\infty}^{1/2},\lVert
V_{n}^{2}-1\rVert_{1}^{1/3}\bigg{\\}}$ (11)

and that the first term $\tfrac{\log n}{n^{1/2}}$ is optimal for the case
$s_{n}^{2}=n$. Haeusler [24, 25] generalized (10) to $2<p<\infty$ and showed
that the first term $L_{p}^{p/(p+1)}$ is exact. Joos [31] proved that the
second term in (10) can be replaced by $\lVert
V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}$, $q\geq 1$. Mourrat [34] improved the second
term in (11) to $(\lVert V_{n}^{2}-1\rVert_{q}+s_{n}^{-2})^{q/(2q+1)}$, $q\geq
1$. Cf. also [26, 31, 23, 35, 15, 17, 19] and references therein for work in
this direction.

###  1.2 Motivation and our contributions

Our paper concerns the $\mathcal{W}_{r}$ convergence rates for the martingale
CLT, $1\leq r\leq 3$.

Let us comment on some weaknesses of the aforementioned $\mathcal{W}_{1}$
bounds.

When the martingale differences are at least $\mathscr{L}^{p}$-integrable,
$2<p\leq 3$, the best $\mathcal{W}_{1}$ rates given by (7), (9), (8) are
typically $n^{1/p-1/2}\geq n^{-1/6}$, leaving a big gap between the rate
$O(n^{-1/2}\log n)$ in the $\mathscr{L}^{\infty}$ case (5), not to mention
that the condition (6) imposed in (9) is too restrictive for general
martingales. Compared to (9), the exponent of $L_{p}$ in (4) is clearly not
optimal, at least for $2<p\leq 3$. But (4) does not assume (6), and it offers
a typical $\mathcal{W}_{1}$ rate $O(n^{(2-p)/(2p+2)})$ which is better than
$n^{-1/6}$ for $p>7/2$. However, the constant $c_{p}$ in (4) is expected to
grow linearly as $p\to\infty$, rendering (4) a useless bound when $p$ is
bigger than $n^{1/2}$.

Notice that all of the results discussed above are $\mathcal{W}_{1}$ bounds.
There are rarely any results on the $\mathcal{W}_{r}$ bounds, $r>1$, in terms
of $L_{p}$ for (general) martingales.

Can we obtain $\mathcal{W}_{r}$ bounds, $r\geq 1$, in terms of the Lyapunov
coefficient (2) for the martingale CLT that generalize all of the previous
results (4), (5), (7), (8), (9)? Further, what are the optimal rates and how
to justify their optimality?

Is it possible to get Wasserstein distance bounds for the CLT so that the
constant coefficents do not blow up as the integrability of the martingale
increases? Such estimates would be important when we have a limited sample
size, and they allow us to exploit the integrability of the martingale to
obtain better rates.

In theory and in applications, there are numerous stochastic processes that do
not fit into any of the $\mathscr{L}^{p}$ category, $2<p<\infty$, cf. [41, 32,
43]. Can we quantify the CLT for martingales with much wider spectrum of
integrability than $\mathscr{L}^{p}$, $p\ll n^{1/2}$?

Motivated by these questions, we will prove the following results.

  1. (1)

We will obtain $\mathcal{W}_{r}$, $1\leq r\leq 3$, convergence rates for
martingales which are Orlicz-integrable. For instance, if

$A:=\frac{1}{n}\sum_{i=1}^{n}\psi(|Y_{i}|)<\infty$ (12)

for appropriate convex function $\psi$ that grows at most polynomially fast,
then

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{\psi}\frac{A\vee
1}{s_{n}}\psi^{-1}(n)+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2},$ (13)

where $\psi^{-1}$ denotes the inverse function of $\psi$.

Our result greatly generalizes the known $\mathcal{W}_{1}$ bounds (4), (7),
(8), (9) for $\mathscr{L}^{p}$ martingales, $2<p\leq 3$. Moreover, we will
prove that both terms $\tfrac{1}{s_{n}}\psi^{-1}(n)$ and $\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}$ in this $\mathcal{W}_{1}$ bound are optimal.

  2. (2)

We will derive Wasserstein bounds whose constant coefficients do not depend on
the integrability of the variables. Take the $\mathcal{W}_{1}$ distance for
example, if (12) holds for $\psi$ in a wide class of convex functions, we show
that

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{A\vee
1}{s_{n}}\psi^{-1}(n)\log n+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.$ (14)

This explains the presence of the $\log n$ term in (5), and it implies that if
the mds is $\mathscr{L}^{\infty}$ bounded, i.e., $M=\max_{i=1}^{n}\lVert
Y_{i}\rVert_{\infty}<\infty$, then

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{M}{s_{n}}\log(e+\frac{s_{n}^{2}}{M^{2}})+\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}.$

The novelty of this result lies not only on the fact that it encompasses an
even larger spectrum (all the way up to $\mathscr{L}^{\infty}$) of
integrability than our first bound (13), but also that it yields a better
bound than (13) when the order of the “integrability” $\psi$ is bigger than
the logarithm of the sample size (i.e., beyond $O(\log n)$).

  3. (3)

Similar bounds for the $\mathcal{W}_{r}$ distances in terms of the Lyapunov
coefficient and $\lVert V_{n}^{2}-1\rVert_{r/2}$, $1<r\leq 3$, will be
established as well.

###  1.3 Structure of the paper

The organization of our paper is as follows.

Subsection 1.4 contains definitions of N-functions and the corresponding
Orlicz norm for random sequences. In Section 2, we present our main
Wasserstein distance bounds (Theorems 5,7, and 8) and the optimality of the
$\mathcal{W}_{1}$ rate (Proposition 10). In Section 3, using Taylor expansion
and Lindeberg’s telescopic sum argument, we will derive $\mathcal{W}_{r}$
bounds in terms of the conditional moments for martingales with $V_{n}^{2}=1$
a.s.. As consequences, $\mathcal{W}_{1},\mathcal{W}_{2}$ bounds (Corollary 15)
will be obtained for martingales that satisfy certain special conditions.

Section 4 is devoted to the proof of our main results. Our proof consists of
the following components. First, we truncate the martingale as Haeusler [25]
and elongate it as Dvoretzky [13] to turn it into a sequence with bounded
increments and $V_{n}^{2}=1$. We will bound the error of this modification in
terms of the Lyapunov conefficient and $(V_{n}^{2}-1)$. (See Section 4.1.)
Then, as a crucial technical step, we use Young’s inequality within
conditional expectations to “decouple” the Lyapunov coefficient and
$\sigma_{i}^{2}(Y)$’s from the conditional moments, and turn the
$\mathcal{W}_{r}$ bound to be an optimization problem over three parameters.
This argument is robust enough for us to deal with martingales with very
flexible integrability. Our another key observation is that there should be
different bounds for the two different scenarios when the martingale is “at
most $\mathscr{L}^{p}$” and “more integrable than $\mathscr{L}^{p}$”. By
possibly sacrificing a small factor (e.g. $\log n$), we can make the constants
of the bound independent of the integrability $\psi$, which leads to bounds of
type (14).

Finally, in Section 5 we construct examples to show that our $\mathcal{W}_{1}$
bound is optimal. We leave some open questions in Section 6.

###  1.4 Preliminaries: N-functions and an Orlicz-norm for sequences

To generalize the notion of $\mathcal{L}^{p}$ integrability, we recall the
definitions of N-function and the corresponding Orlicz-norm in this
subsection.

######  Definition 2.

A convex function $\psi:[0,\infty)\to[0,\infty)$ is called an N-function if it
satisfies $\lim_{x\to 0}\psi(x)/x=0$, $\lim_{x\to\infty}\psi(x)/x=\infty$, and
$\psi(x)>0$ for $x>0$. For two functions
$\psi_{1},\psi_{2}\in[0,\infty)^{[0,\infty)}$, we write

$\psi_{1}\preccurlyeq\psi_{2}\quad\text{ or
}\quad\psi_{2}\succcurlyeq\psi_{1}$

if $\frac{\psi_{2}(x)}{\psi_{1}(x)}$ is non-decreasing on $(0,\infty)$.

Note that every N-function $\psi$ satisfies $\psi\succcurlyeq x$. See [1].

Denote the Fenchel-Legendre transform of $\psi$ by
$\psi_{*}\in[0,\infty)^{[0,\infty)}$, i.e.,

$\psi_{*}(x)=\sup\\{xy-\psi(y):y\in\mathbb{R}\\}\quad\text{ for }x\geq 0.$

Then $\psi_{*}$ is still an N-function.Young’s inequality states that

$xy\leq\psi(x)+\psi_{*}(y)\quad\text{ for all }x,y\geq 0.$ (15)

Another useful relation between the pair $\psi,\psi_{*}$ is

$x\leq\psi^{-1}(x)\psi_{*}^{-1}(x)\leq 2x,\quad\forall x\in[0,\infty).$ (16)

See [1] for a proof and for more properties of N-functions.

######  Definition 3.

For any N-function $\psi$ and $n\in\mathbb{N}$, the
$\mathscr{L}^{\psi}$-Orlicz norm of a random sequence
$Y=\\{Y_{m}\\}_{m=1}^{n}$ with length $n$ is defined by

$\lVert
Y\rVert_{\psi}:=\inf\big{\\{}c>0:\frac{1}{n}\sum_{i=1}^{n}E\left[\psi(|Y_{i}|/c)\right]\leq
1\big{\\}}.$ (17)

In particular, when $n=1$, i.e., $Y=Y_{1}$ is a single random variable, we
still write

$\lVert Y_{1}\rVert_{\psi}=\inf\\{c>0:E[\psi(|Y_{1}|/c)]\leq 1\\}.$

When $\psi(x)=x^{p}$, $p\geq 1$, we simply write $\lVert Y\rVert_{\psi}$ as
$\lVert Y\rVert_{p}$. Notice that

$\displaystyle\lVert Y\rVert_{p}$
$\displaystyle=\left(\frac{1}{n}\sum_{i=1}^{n}E[|Y_{i}|^{p}]\right)^{1/p},$
(18) $\displaystyle\text{and }\quad\lim_{p\to\infty}\lVert Y\rVert_{p}$
$\displaystyle=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}=:\lVert
Y\rVert_{\infty}.$ (19)

######  Remark 4.

  1. (a)

Using the property $\psi(\lambda x)\geq\lambda\psi(x)$ for $\lambda\geq 1$ for
N-functions, it is easily seen that the integrability condition (12) implies
$\lVert Y\rVert_{\psi}\leq A\vee 1$.

  2. (b)

Since this paper concerns square-integrable martingales, we are only
interested in sup-quadratic N-functions $\psi\succcurlyeq x^{2}$. For
instance, $x^{2}\log(x+1)$, $x^{p}$ (with $p\geq 2$), $e^{x}-x-1$,
$\exp(x^{\beta})-1$, $\exp\left(\ln(x+1)^{\beta}\right)-1$ (with $\beta>1$)
are among such N-functions.

  3. (c)

The meaning of our notation $\lVert Y\rVert_{p}$ is different from some work
in the literature, e.g. [5, 19] where it means $\max_{i=1}^{n}\lVert
Y_{i}\rVert_{p}$. Note that $\lVert Y\rVert_{\psi}\leq\max_{i=1}^{n}\lVert
Y_{i}\rVert_{\psi}$.

  4. (d)

Removing the $\tfrac{1}{n}$ in (18) only changes a multiplicative factor of
$\lVert Y\rVert_{p}$. However, for general $\lVert\cdot\rVert_{\psi}$, the
presence of $\tfrac{1}{n}$ in (17) is crucial for the definition–removing it
would drastically change the meaning of the norm.

##  2 Main results

Recall $\lVert Y\rVert_{\infty}$ in (19). For the mds $\\{Y_{i}\\}_{i=1}^{n}$
and an N-function $\psi$, we generalize the notation $L_{p}$ in (2) to be

$L_{\psi}=\frac{1}{s_{n}}\lVert Y\rVert_{\psi}\psi^{-1}(n),\quad
L_{\infty}=\frac{1}{s_{n}}\lVert Y\rVert_{\infty},$ (20)

where $\psi^{-1}$ denotes the inverse function of $\psi$.

Our first main results are two $\mathcal{W}_{1}$ bounds that generalize (4),
(5), (7), (8), (9).

######  Theorem 5 ($\mathcal{W}_{1}$ bounds).

Let $\\{Y_{m}\\}_{m=1}^{n}$ be a martingale difference sequence. Recall the
notations $X_{n},V_{n}^{2},L_{\psi},L_{\infty},\preccurlyeq,\succcurlyeq$ in
(1), (20), and Definition 2.

  1. (i)

For any N-function $\psi\succcurlyeq x^{2}$,

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim
L_{\psi}\log\left(e+L_{\psi}^{-2}\right)+\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}.$ (21)

  2. (ii)

For any N-function $\psi$ with $x^{2}\preccurlyeq\psi\preccurlyeq x^{p}$,
$p>2$,

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim pL_{\psi}+\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}.$ (22)

As consequences, for any $p>2$,

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\min\left\\{p,\,\log(e+L_{p}^{-2})\right\\}L_{p}+\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}.$

In particular, for the $\mathscr{L}^{\infty}$ case, by (19),

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim
L_{\infty}\log\left(e+L_{\infty}^{-2}\right)+\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}.$

######  Remark 6.

  1. (a)

For any sup-quadratic N-function $\psi\succcurlyeq x^{2}$, by Lemma A.1,

$s_{n}=\lVert Y\rVert_{2}n^{1/2}\lesssim_{\psi(1)}\lVert
Y\rVert_{\psi}n^{1/2}.$

Hence, by the definition of $L_{\psi}$ in (20),

$L_{\psi}\gtrsim_{\psi(1)}\psi^{-1}(n)n^{-1/2}\gtrsim_{\psi(1)}\psi^{-1}(1)n^{-1/2}.$

Thus, $\log(e+L_{\psi}^{-2})$ in Theorem 5(i) is dominated by $\log(n)$.

  2. (b)

Both bounds (21) and (22) in Theorem 5 have their strengths and weaknesses.

Apparently, the second bound (22) gives better rates for mds which are at most
polynomially integrable. E.g., for the “barely more than square integrable”
case $\psi=x^{2}\log(x+1)$, (22) yields $\frac{1}{s_{n}}(\frac{n}{\log
n})^{1/2}(\log\log n)\lVert Y\rVert_{\psi}$ as the first term in the bound.
However, (22) becomes a trivial bound for $p\gg n^{1/2}$.

Although the first bound (21) is seemingly $\log(n)$ factor worse than the
latter, it is applicable to martingales with more general integrability. For
instance, if the martingale is exponentially integrable, taking
$\psi=e^{x}-x-1$, Theorem 5(i) yields
$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\lVert Y\rVert_{\psi}(\log
n)^{2}/s_{n}+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$, where the first term is
better than the $O(n^{1/p}/s_{n})$ rates offered by (22) for any $p>0$.

  3. (c)

In some sense, the term $\lVert V_{n}^{2}-1\rVert$ quantifies the extent of
decorrelation of the process. For independent sequences, $(V_{n}^{2}-1)$ is
$0$. In general, $(V_{n}^{2}-1)$ could converge to $0$ arbitrarily fast,
depending on how decorrelated the process is.

  4. (d)

The term $(V_{n}^{2}-1)$ is essential for bounds of type

$\mathcal{W}_{r}(X_{n},\mathcal{N})\leq C_{1}L_{\psi}^{a}+C_{2}\lVert
V_{n}^{2}-1\rVert_{c}^{b},$ (23)

i.e., we cannot allow $C_{2}$ to be $0$. For example, let $B$ be such that
$P(B=1\pm\tfrac{1}{2})=\tfrac{1}{2}$ and let $(\xi_{i})_{i=1}^{n}$ be i.i.d.
$\mathcal{N}(0,1)$ variables. Define $Y_{i}=(B/n)^{1/2}\xi_{i}$. Then
$\lim_{n\to\infty}\mathcal{W}_{r}(\sum_{i=1}^{n}Y_{i},\mathcal{N})=\mathcal{W}_{r}(\sqrt{B}\xi_{1},\mathcal{N})\neq
0$. Whereas, for $p>2$, $L_{p}=O(n^{1/p-1/2})\to 0$ as $n\to\infty$. Thus
$C_{2}\neq 0$.

Our next main results concern the $\mathcal{W}_{2}$ and $\mathcal{W}_{3}$
convergence rates.

######  Theorem 7 ($\mathcal{W}_{2}$ bounds).

Let $\\{Y_{m}\\}_{m=1}^{n}$ be a martingale difference sequence.

  1. (i)

For any N-function $\psi\preccurlyeq x^{3}$ such that $x\mapsto\psi(\sqrt{x})$
is convex,

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim L_{\psi}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2}.$

  2. (ii)

For any N-function $\psi\succcurlyeq x^{3}$,

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim_{\psi(1)}L_{\psi}+\frac{\lVert
Y\rVert_{\psi}}{s_{n}}\left[ng^{-1}(\psi^{-1}(n)^{2})\right]^{1/4}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2},$

where $g$ denotes the inverse function of $x\mapsto\tfrac{\psi(x)}{x}$.

As consequences, for $p\in(2,\infty)$,

$\displaystyle\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\left\\{\begin{array}[]{lr}L_{p}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2}&\text{ when }p\in(2,3],\\\
n^{1/4+1/(2p^{2}-2p)}s_{n}^{-1}\lVert Y\rVert_{p}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2}&\text{ when }p>3.\end{array}\right.$

In particular, taking $p\to\infty$,

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\frac{n^{1/4}}{s_{n}}\lVert
Y\rVert_{\infty}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}.$ (24)

######  Theorem 8 ($\mathcal{W}_{3}$ bound).

Let $\\{Y_{m}\\}_{m=1}^{n}$ be a martingale difference sequence. Then

$\mathcal{W}_{3}(X_{n},\mathcal{N})\lesssim L_{3}+\lVert
V_{n}^{2}-1\rVert_{3/2}^{1/2}.$

######  Remark 9.

For the ease of our presentation, we only present results in terms of the
$\mathcal{W}_{r}$ metrics, $r\in\\{1,2,3\\}$. Readers can use the
interpolation inequality

$\mathcal{W}_{r}(X_{n},\mathcal{N})\leq\mathcal{W}_{j}(X_{n},\mathcal{N})^{j(k-r)/(k-j)}\mathcal{W}_{k}(X_{n},\mathcal{N})^{k(r-j)/(k-j)},\quad
j<r<k,$

to easily infer other $\mathcal{W}_{r}$ bounds, $r\in(1,3)$.

The following proposition justifies that the terms $L_{\psi}$ and the exponent
$1/2$ of $\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$ within the $\mathcal{W}_{1}$
bound (22) are optimal.

######  Proposition 10.

(Optimality of the $\mathcal{W}_{1}$ rates).

  1. (1)

For any $p>2$, there exists a constant $N$ depending on $p$ such that for any
$n>N$, we can find a martingale $\\{X_{i}\\}_{i=1}^{n}$ which satisfies

     * •

$V_{n}^{2}=1$, a.s.;

     * •

$L_{p}\lesssim_{p}(\log n)^{-1}$;

     * •

$\mathcal{W}_{1}(X_{n},\mathcal{N})\asymp_{p}L_{p}$.

  2. (2)

For any $n\geq 2$, there exists a martingale $\\{X_{i}\\}_{i=1}^{n}$ that
satisfies

     * •

$\lVert V_{n}^{2}-1\rVert_{1/2}\lesssim(\log n)^{-2}$;

     * •

$\mathcal{W}_{1}(X_{n},\mathcal{N})\asymp\lVert
V_{n}^{2}-1\rVert_{1/2}^{1/2}$.

##  3 Martingales with $V_{n}^{2}=1$

In this section we will derive $\mathcal{W}_{r}$ bounds, $r\in\\{1,2,3\\}$,
for martingales with $V_{n}^{2}=1$. As in [39, 18, 11, 19], we will use
Lindeberg’s argument and Taylor expansion to bound the Wasserstein distance
with a sum involving the third order conditional moments. Our derivation of
the $\mathcal{W}_{r}$ bounds for $r>1$ also relies on an observation of Rio
[38] that relates the Wasserstein distances to Zolotarev’s ideal metric (25).

Note that [39, 18, 19] use Stein’s method to express the $\mathcal{W}_{1}$
bound in terms of derivatives of Stein’s equation. But as observed in [11],
Gaussian smoothing can already provide us enough regularity to do Taylor
expansions. We follow [11] in this respect.

###  3.1  Wasserstein distance bounds via Lindeberg’s argument

For any $n$-th order differentiable function $f$ and $\gamma\in(0,1]$, denote
its $(n,\gamma)$-Hölder constant by

$[f]_{n,\gamma}:=\sup_{x,y:x\neq
y}\frac{f^{(n)}(x)-f^{(n)}(y)}{|x-y|^{\gamma}}.$

We say that $f\in C^{n,\gamma}$ if $[f]_{n,\gamma}<\infty$.

######  Definition 11.

Suppose $r>1$, and $\mu$, $\nu$ are probability measures on $\mathbb{R}$. Let
$\ell\in\mathbb{N}$ and $\gamma\in(0,1]$ be the unique numbers such that
$r=\ell+\gamma$. We let $\Lambda_{r}=\\{f\in
C^{\ell,\gamma}:[f]_{\ell,\gamma}\leq 1\\}$. The Zolotarev distance
$\mathcal{Z}_{r}$ is defined by

$\mathcal{Z}_{r}(\mu,\nu)=\sup\left\\{\int_{\mathbb{R}}f\mathrm{d}\mu-\int_{\mathbb{R}}f\mathrm{d}\nu:f\in\Lambda_{r}\right\\}.$

Recall that by the Kantorovich-Rubinstein Theorem,
$\mathcal{W}_{r}(\mu,\nu)=\mathcal{Z}_{r}(\mu,\nu)$ for $r=1$. When $r>1$, it
is shown by Rio [38, Theorem 3.1] that

$\mathcal{W}_{r}(\mu,\nu)\lesssim_{r}\mathcal{Z}_{r}(\mu,\nu)^{1/r}.$ (25)

For $\sigma>0$ and any function $f\in\Lambda_{r}$, let

$f_{\sigma}(x):=E[f(x+\sigma\mathcal{N})],$ (26)

where $\mathcal{N}$ denotes a standard normal random variable. Direct
integration by parts would yield the following regularity estimate for the
Gaussian smoothing $f_{\sigma}$ which is a special case of [10, Lemma 6.1].

######  Lemma 12.

[10, Lemma 6.1] For $\sigma>0$ and $f\in\Lambda_{r}$, $r\in\mathbb{N}$, let
$f_{\sigma}$ be as in (26). Then

$\lVert f_{\sigma}^{(k)}\rVert_{\infty}\leq C_{k+1-r}\sigma^{r-k},\quad\forall
k\geq r,$

where $C_{n}=\int_{\mathbb{R}}\lvert u\phi^{(n)}(u)\rvert\mathrm{d}u$, and
$\phi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ is the standard normal density.

For the reader’s convenience, we include a proof of Lemma 12 in the Appendix.

Recall the notations $\sigma_{m}(Y),X_{m}$ in (1). For any random variable
$U$, we write the expectation conditioning on $\mathcal{F}_{i-1}$ as $E_{i}$,
i.e.,

$E_{i}[U]:=E[U|\mathcal{F}_{i-1}].$ (27)

######  Proposition 13.

Let $Y=\\{Y_{m}\\}_{m=1}^{n}$, $n\geq 2$, be a martingale difference sequence
and let $\beta>0$. Assume that $\sum_{i=1}^{n}\sigma_{i}^{2}(Y)=s_{n}^{2}$
almost surely. For any $\beta>0$, set

$\lambda_{m}^{2}=\lambda_{m}^{2}(Y,\beta):=\sum_{i=m+1}^{n}\sigma_{i}^{2}(Y)+\beta^{2}s_{n}^{2}.$

Then, for any $x^{3}\succcurlyeq\psi\in[0,\infty)^{[0,\infty)}$ such that
$x\mapsto\psi(\sqrt{x})$ is convex,

$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$
$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\min\left\\{\frac{E_{i}[|Y_{i}|^{3}]}{\lambda_{i}^{3-r}},\frac{\lambda_{i}^{r}E_{i}\psi(|Y_{i}|)}{\psi(\lambda_{i})}\right\\},\,r\in\\{1,2\\},$
(28) $\displaystyle\mathcal{Z}_{3}(X_{n},\mathcal{N})$
$\displaystyle\lesssim\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|Y_{i}|^{3}].$ (29)

###### Proof of Proposition 13:.

Without loss of generality , assume $s_{n}=1$.

Let $\xi,\mathcal{N}$ be random variables with distributions
$\mathcal{N}(0,\beta^{2})$ and $\mathcal{N}(0,1)$, so that the triple
$(\xi,\mathcal{N},\\{X_{i}\\}_{i=1}^{n})$ are independent. For
$h\in\Lambda_{r}$, $r\in\\{1,2\\}$, and any random variable $U$ which is
independent of $\xi$,

$\displaystyle\lvert
E[h(U+\xi)-h(U)]\rvert=\Bigr{\lvert}E[h(U+\xi)-\sum_{i=0}^{r-1}h^{(i)}(U)\xi^{i}]\Bigl{\rvert}\leq\tfrac{1}{r!}E[|\xi|^{r}].$

The triangle inequality and (25) then yields, for $r\in\\{1,2\\}$,

$\lvert\mathcal{Z}_{r}(X_{n},\mathcal{N})-\mathcal{Z}_{r}(X_{n}+\xi,\mathcal{N}+\xi)\rvert\lesssim\beta^{r}.$
(30)

In what follows we will use Lindeberg’s argument to derive bounds for
$\mathcal{Z}_{r}(X_{n}+\xi,\mathcal{N}+\xi)$, $r\in\\{1,2,3\\}$. Recall
notations in (1), (27).

Note that $\lambda_{m}$ is $\mathcal{F}_{m-1}$-measurable. Recall the function
$h_{\sigma}$ in (26). Recall the operation $E_{i}$ in (27). For
$h\in\Lambda_{r}$, we consider the following telescopic sum:

$\displaystyle E[h(X_{n}+\xi)-h(\mathcal{N}+\xi)]$
$\displaystyle=\sum_{i=1}^{n}E[h(X_{i}+\lambda_{i}\mathcal{N})-h(X_{i-1}+\lambda_{i-1}\mathcal{N})]=E\sum_{i=1}^{n}D_{i},$

where
$D_{i}:=E_{i}[h_{\lambda_{i}}(X_{i})-h_{\lambda_{i}}(X_{i-1}+\sigma_{i}\mathcal{N})]$.

Note that

$D_{i}=E_{i}\int_{\sigma_{i}\mathcal{N}}^{Y_{i}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t.$
(31)

Hence, for $h\in\Lambda_{r}$, $r\in\\{1,2,3\\}$, using the fact
$\sigma_{i}^{3}\leq E_{i}[\lvert Y_{i}\rvert^{3}]$, we have

$|D_{i}|\lesssim E_{i}[|Y_{i}|^{3}]\lVert
h_{\lambda_{i}}^{(3)}\rVert_{\infty}\stackrel{{\scriptstyle
Lemma~{}\ref{lem:smoothing}}}{{\lesssim}}E_{i}[|Y_{i}|^{3}]\lambda_{i}^{r-3}.$
(32)

When $r=3$, notice that $\beta$ is irrelevant . So, letting $\beta\downarrow
0$ we immediately get (29). It remains to consider $r\in\\{1,2\\}$.

Using the fact that for $f\in C^{1,1}$,

$\displaystyle\Bigr{\lvert}\int_{0}^{y}\int_{0}^{t}f^{\prime\prime}(x+s)-f^{\prime\prime}(x)\mathrm{d}s\mathrm{d}t\Bigl{\rvert}\leq\min\left\\{\tfrac{1}{6}|y|^{3}\lVert
f^{(3)}\rVert_{\infty},\,y^{2}\lVert
f^{\prime\prime}\rVert_{\infty}\right\\},$

we get, for any event $A_{i}$ and any $h\in\Lambda_{r}$, $r\in\\{1,2\\}$,

$\displaystyle\mathrm{I}:$
$\displaystyle=\Bigr{\lvert}E_{i}\int_{0}^{Y_{i}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t\Bigl{\rvert}$
$\displaystyle\leq E_{i}\left[\tfrac{1}{6}|Y_{i}|^{3}\lVert
h_{\lambda_{i}}^{(3)}\rVert_{\infty}\mathbbm{1}_{A_{i}^{c}}+Y_{i}^{2}\lVert
h_{\lambda_{i}}^{\prime\prime}\rVert_{\infty}\mathbbm{1}_{A_{i}}\right]$
$\displaystyle\stackrel{{\scriptstyle
Lemma~{}\ref{lem:smoothing}}}{{\lesssim}}E_{i}\left[|Y_{i}|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{A_{i}^{c}}+Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{A_{i}}\right],$

and similarly

$\displaystyle\mathrm{II}:$
$\displaystyle=\Bigr{\lvert}E_{i}\int_{0}^{\sigma_{i}\mathcal{N}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t\Bigl{\rvert}$
$\displaystyle\lesssim
E_{i}\left[\sigma^{3}\lambda_{i}^{r-3}\mathbbm{1}_{A_{i}^{c}}+\sigma_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{A_{i}}\right].$

Further, since $\psi\preccurlyeq x^{3}$, we have
$s^{3}/t^{3}\leq\psi(s)/\psi(t)$ for $0<s\leq t$. Hence

$|Y_{i}|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}\leq\lambda_{i}^{r}\frac{\psi(|Y_{i}|)}{\psi(\lambda_{i})}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}.$

Since $\psi\succcurlyeq x^{2}$, we have
$s^{2}/t^{2}\leq\psi_{1}(s)/\psi_{1}(t)$ for $0<t\leq s$. Thus

$Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}}\leq\lambda_{i}^{r}\frac{\psi(|Y_{i}|)}{\psi(\lambda_{i})}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}},$

and so, for $r\in\\{1,2\\}$,

$\displaystyle\mathrm{I}\lesssim
E_{i}\left[|Y_{i}|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}+Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}}\right]\lesssim\frac{\lambda_{i}^{r}E_{i}[\psi(|Y_{i}|)]}{\psi(\lambda_{i})}.$

Similarly, for $h\in\Lambda_{r}$, $r\in\\{1,2\\}$,

$\mathrm{II}\lesssim
E_{i}\left[\sigma^{3}\lambda_{i}^{r-3}\mathbbm{1}_{\sigma_{i}<\lambda_{i}}+\sigma_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{\sigma_{i}\geq\lambda_{i}}\right]\lesssim\frac{\lambda_{i}^{r}\psi(\sigma_{i})}{\psi(\lambda_{i})}.$

Since $x\mapsto\psi(\sqrt{x})$ is convex, by Jensen’s inequality we know that

$\psi(\sigma_{i})\leq E_{i}[\psi(|Y_{i}|)].$ (33)

Thus, we obtain, for $h\in\Lambda_{r}$, $r\in\\{1,2\\}$,

$|D_{i}|\stackrel{{\scriptstyle\eqref{eq:di-
bound}}}{{\leq}}\mathrm{I}+\mathrm{II}\lesssim\frac{\lambda_{i}^{r}E_{i}[\psi(|Y_{i}|)]}{\psi(|\lambda_{i}|)}.$

This inequality, together with (32), (30) and (25), yields the Proposition. ∎

######  Remark 14.

If there exists $1\leq j\leq n$ such that, given $\mathcal{F}_{j-1}$, the
distribution of $Y_{j}$ is the normal $\mathcal{N}(0,\sigma_{j}^{2})$, then
the $j$-th summand in (28) and (29) can be removed. That is, the summations in
(28) and (29) can both be replaced by

$\sum_{i=1,i\neq j}^{n}.$

Indeed, in this case, $X_{j}$ and $X_{j-1}+\sigma_{j}\mathcal{N}$ are
identically distributed conditioning on $\mathcal{F}_{j-1}$. Thus $D_{j}=0$.

###  3.2 The rates of some special cases

From Proposition 13 we can obtain Wasserstein distance bounds for some special
cases (with $V_{n}^{2}=1$). See Corollary 15 below. These cases usually yield
better rates than the typical cases. They are not only of interest on their
own right, but can also serve as important references when we construct
counterexamples.

The $\mathcal{W}_{1}$ bounds within Corollary 15 can be considered
generalizations of some of the results in [39, Corollaries 2.2,2.3] and [19,
Corollaries 2.5,2.6,2.7] to the $\mathscr{L}^{\psi}$ integrable cases.

######  Corollary 15.

Assume that $V_{n}^{2}=1$ almost surely. Let $\psi$ be an N-function such that
$\psi\preccurlyeq x^{3}$ and $x\mapsto\psi(\sqrt{x})$ is convex. Then the
following statements hold.

  1. (i)

$\mathcal{W}_{r}(X_{n},\mathcal{N})\lesssim L_{\psi}$ for $r\in\\{1,2\\}$.

  2. (ii)

If there exists $\sigma>0$ such that

$P(\sigma_{i}\geq\sigma\,\text{ for all }1\leq i\leq n)=1,$

then, writing $M_{\psi}=\max_{i=1}^{n}E[\psi(|Y_{i}|)]$ (and write $M_{\psi}$
as $M_{3}$ when $\psi=x^{3}$),

     * •

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\left\\{\begin{array}[]{lr}\frac{M_{3}}{\sigma^{2}s_{n}}\log
n&\text{ when }\psi=x^{3},\\\
(3-p)^{-1}\frac{M_{\psi}s_{n}^{2}}{\sigma^{2}\psi(s_{n})}&\text{ when
}\psi\preccurlyeq x^{p},p\in(2,3);\end{array}\right.$.

     * •

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\frac{M_{\psi}^{1/2}s_{n}}{\sigma\psi(s_{n})^{1/2}}$.

  3. (iii)

If there exists a constant $\theta>0$ such that almost surely,

$E[\psi(|Y_{i}|)|\mathcal{F}_{i-1}]\leq\theta\sigma_{i}^{2}\quad\forall
i=1,\ldots,n,$

then

     * •

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{\theta}\left\\{\begin{array}[]{lr}\frac{1}{s_{n}}\log(e+s_{n})&\text{
when }\psi=x^{3},\\\ \frac{s_{n}^{2}}{\psi(s_{n})}&\text{ when
}\psi\preccurlyeq x^{p},p\in(2,3);\end{array}\right.$

     * •

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim_{\theta}\frac{s_{n}}{\psi(s_{n})^{1/2}}$.

In particular, when $\theta=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}<\infty$,
we have

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{\theta}{s_{n}}\log(e+s_{n})\,\text{
and
}\,\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim(\tfrac{\theta}{s_{n}})^{1/2}.$

###### Proof.

(i) For the ease of notation we write $\theta:=\lVert Y\rVert_{\psi}$. Note
that $x\mapsto\psi(x/\theta)$ is still an N-function that satisfies conditions
of Proposition 13. Recall the conditional expectation $E_{i}$ in (27). By
Proposition 13,

$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$
$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{\lambda_{i}^{r}E_{i}[\psi(|Y_{i}|/\theta)]}{\psi(\lambda_{i}/\theta)}$
$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{(\beta
s_{n})^{r}E_{i}[\psi(|Y_{i}|/\theta)]}{\psi(\beta s_{n}/\theta)}$
$\displaystyle\lesssim\beta^{r}+\frac{n\beta^{r}}{\psi(\beta
s_{n}/\theta)}\quad\text{ for any }\beta>0,r\in\\{1,2\\},$

where in the second inequality we used the fact that $\lambda_{i}\geq\beta
s_{n}$ for all $i$ and that $x\mapsto\frac{x^{r}}{\psi(x)}$ is decreasing.
Taking $\beta=\frac{\theta}{s_{n}}\psi^{-1}(n)$, the $\mathcal{Z}_{1}$,
$\mathcal{Z}_{2}$ bounds are proved.

(ii) Recall $\lambda_{i}$ in Proposition 13. When $\sigma_{i}\geq\sigma$ for
all $i$ we have

$\lambda_{i}^{2}\geq(n-i)\sigma^{2}+\beta^{2}s_{n}^{2},\quad\forall 1\leq
i\leq n.$

Thus, by Proposition 13, we get, for any $\beta\geq 2\sigma/s_{n}$,
$x^{r}\preccurlyeq\psi\preccurlyeq x^{3}$, $r\in\\{1,2\\}$,

$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$
$\displaystyle\lesssim\beta^{r}+\frac{M_{\psi}}{s_{n}^{r}}\sum_{i=1}^{n}\tfrac{[(n-i)\sigma^{2}+\beta^{2}s_{n}^{2}]^{r/2}}{\psi(\sqrt{(n-i)\sigma^{2}+\beta^{2}s_{n}^{2}})}$
$\displaystyle\lesssim\beta^{r}+\frac{M_{\psi}}{s_{n}^{r}\sigma^{2}}\int_{\sigma}^{\sqrt{n}\sigma+\beta
s_{n}}\frac{t^{r+1}}{\psi(t)}\mathrm{d}t.$ (34)

The first $\mathcal{Z}_{1}$ bound follows when $r=1$, $\psi(x)=x^{3}$,
$\beta=2M_{3}/(s_{n}\sigma^{2})$.

Now consider the case $x^{r}\preccurlyeq\psi\preccurlyeq x^{p}$,
$p\in(2,r+2)\cap(2,3]$, $r\in\\{1,2\\}$. Since $s_{n}^{2}\geq n\sigma$ and
$x\mapsto x^{p}/\psi(x)$ is increasing, we get, for any
$\beta\in[\tfrac{2\sigma}{s_{n}},1]$,

$\displaystyle\int_{\sigma}^{\sqrt{n}\sigma+\beta
s_{n}}\frac{t^{p}}{\psi(t)}t^{r+1-p}\mathrm{d}t$
$\displaystyle\leq\frac{(2s_{n})^{p}}{\psi(2s_{n})}\int_{0}^{2s_{n}}t^{r+1-p}\mathrm{d}t$
$\displaystyle\leq\frac{1}{r+2-p}\frac{(2s_{n})^{r+2}}{\psi(2s_{n})}$
$\displaystyle\leq\frac{1}{r+2-p}\frac{2^{r}s_{n}^{r+2}}{\psi(s_{n})}.$

Taking $\beta$ such that
$\beta^{r}=C\frac{M_{\psi}}{\sigma^{2}}\frac{s_{n}^{2}}{\psi(s_{n})}$ for
appropriate constant $C>0$ and recalling (3.2), we get (the case
$x^{r}\preccurlyeq\psi\preccurlyeq x^{p}$, $p\in(2,r+2)\cap(2,3]$)

$\mathcal{Z}_{r}(X_{n},\mathcal{N})\lesssim\frac{1}{r+2-p}\beta^{r}.$

the rest of the $\mathcal{W}_{r}$ bounds in (ii) follow. Note that this is a
trivial inequality (since $\mathcal{W}_{r}(X_{n},\mathcal{N})\lesssim 1$) if
$\beta>1$. If $\beta<1$, it remains to justify that such a choice of $\beta$
satisfies $\beta\geq 2\sigma/s_{n}$. Indeed, since $\psi\succcurlyeq x^{2}$
and $\sigma_{i}\geq\sigma$, we have, for $1\leq i\leq n$,

$M_{\psi}\geq\frac{\psi(\sigma/2)}{(\sigma/2)^{2}}E[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|\geq\sigma/2}]\gtrsim\psi(\sigma/2)$

and so, for $r\in\\{1,2\\}$,

$\beta
s_{n}\geq\beta^{r}s_{n}\gtrsim\frac{\psi(\sigma/2)s_{n}^{3}}{(\sigma/2)^{2}\psi(s_{n})}\gtrsim\sigma,$

where in the last inequality we used the fact that $\psi\lesssim x^{3}$.

(iii) First, notice that there exists $\gamma>0$ such that
$\sigma_{i}<\gamma$, $\forall 1\leq i\leq n$. Indeed,

$\displaystyle\frac{3}{4}\sigma_{i}^{2}$ $\displaystyle\leq
E_{i}[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|>\sigma_{i}/2}]\leq
E_{i}[\psi(|Y_{i}|)]/g(\tfrac{\sigma_{i}}{2})\leq\theta\sigma_{i}^{2}/g(\tfrac{\sigma_{i}}{2})$

where $g(x):=\psi(x)/x^{2}$ is an increasing function on $(0,\infty)$. Thus

$\sigma_{i}\leq 2g^{-1}(\tfrac{4\theta}{3})=:\gamma.$

Next, by Proposition 13 and $E_{i}[\psi(|Y_{i}|)]\leq\theta\sigma_{i}^{2}$, we
get

$\mathcal{Z}_{r}(X_{n},\mathcal{N})\lesssim\beta^{r}+\frac{\theta}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{\lambda_{i}^{r}\sigma_{i}^{2}}{\psi(\lambda_{i})}.$

Define a sequence of stopping times $\\{\tau_{t}\\}_{0\leq t\leq 1}$ by

$\tau(t)=\tau_{t}:=\inf\\{m\geq 1:\sum_{i=1}^{m}\sigma_{i}^{2}\geq t\\}.$ (35)

Clearly, $\tau_{t}=m$ if and only if
$t\in(\sum_{i=1}^{m-1}\sigma_{i}^{2},\sum_{i=1}^{m}\sigma_{i}^{2}]$. Moreover,

$\lambda_{\tau_{t}}^{2}=s_{n}^{2}(1+\beta^{2})-\sum_{i=1}^{\tau_{t}}\sigma_{i}^{2}\geq
s_{n}^{2}(1+\beta^{2})-t-\sigma_{\tau_{t}}^{2}\geq
s_{n}^{2}(1+\beta^{2})-t-\gamma^{2}.$

Thus we get, for any $\beta\in[\frac{\gamma}{s_{n}},1]$, $r\in\\{1,2\\}$,

$\displaystyle\sum_{i=1}^{n}\frac{\lambda_{i}^{r}\sigma_{i}^{2}}{\psi(\lambda_{i})}=\int_{0}^{s_{n}^{2}}\frac{\lambda_{\tau_{t}}^{r}}{\psi(\lambda_{\tau_{t}})}\mathrm{d}t\leq\int_{s_{n}^{2}\beta^{2}-\gamma^{2}}^{(1+\beta^{2})s_{n}^{2}}\frac{t^{r/2}}{\psi(t^{1/2})}\mathrm{d}t\leq
2\int_{\sqrt{s_{n}^{2}\beta^{2}-\gamma^{2}}}^{\sqrt{2}s_{n}}\frac{t^{r+1}}{\psi(t)}\mathrm{d}t,$

where in the first inequality we used the fact that $x\mapsto x/\psi(x)$ is
decreasing. When $\psi=x^{3}$, taking $\beta=2\gamma/s_{n}$, the desired
$\mathcal{W}_{1}$ bound follows. For the case $\psi\preccurlyeq x^{p}$,
$2<p<3$, the bound of this integral can be handled the same way as in (ii).
Then the $\mathcal{W}_{1}$ bounds in (iii) are proved.

It remains to prove the $\mathcal{Z}_{2}$ bound. By the inequality above, for
any $\beta\in[\frac{\gamma}{s_{n}},1]$,

$\displaystyle\mathcal{Z}_{2}(X_{n},\mathcal{N})\lesssim\beta^{2}+\frac{\theta}{s_{n}^{2}}\int_{0}^{\sqrt{2}s_{n}}\frac{t^{3}}{\psi(t)}\mathrm{d}t\lesssim\beta^{2}+\frac{\theta
s_{n}^{2}}{\psi(s_{n})},$

where in the last inequality we used the fact that $x\mapsto x^{3}/\psi$ is
increasing. Taking
$\beta^{2}=\frac{s_{n}^{2}\psi(\gamma)}{\psi(s_{n})\gamma^{2}}$, the
$\mathcal{Z}_{2}$ bound in (iii) follows. ∎

##  4 Proofs of the main $\mathcal{W}_{r}$ bounds

This section is devoted to the proofs of Theorems 5, 7, and 8.

Note that the Taylor expansion results (Proposition 13), which are in terms of
the second and third conditional moments, suit exactly martingales (with
$V_{n}^{2}=1$) with integrability between $\mathscr{L}^{2}$ and
$\mathscr{L}^{3}$, and so not surprisingly, the
$\mathcal{W}_{1},\mathcal{W}_{2}$ bounds of Corollary 15 for $\mathscr{L}^{p}$
martingales, $p\in(2,3]\cup\\{\infty\\}$, followed quite easily from
Proposition 13. However, to obtain Wasserstein bounds for martingales with
more general integrability, we need new insights.

In Subsection 4.1, we will modify the mds into a bounded sequence with
deterministic total conditional variance $V_{n}^{2}$. To this end, we first
truncate the martingale into a bounded sequence with $V_{n}^{2}\leq 1$, and
then lengthen it to have $V_{n}^{2}=1$. Such tricks of elongation and
truncation of martingales were already used by Bolthausen [5] (the idea goes
back to Dvoretzky [13]) and Haeusler [25] in the study of the Kolmogorov
distance of the martingale CLT. The main result of Subsection 4.1 is a control
of the error due to this modification in terms of $L_{\psi}$ and
$(V_{n}^{2}-1)$. See Proposition 17. As an application, we prove the
$\mathcal{W}_{3}$ bound in Theorem 8.

In Subsections 4.2 and 4.3, we prove the $\mathcal{W}_{1},\mathcal{W}_{2}$
estimates in Theorems 5 and 7 by bounding the corresponding metrics for the
modified martingale. A crucial step of our method is to use Young’s inequality
inside the conditional expectations to “decouple” the Lyapunov coefficient
$L_{\psi}$ and the conditional variances $\sigma_{i}^{2}$ from the summation
within Proposition 13. This will turn the $\mathcal{W}_{r}$ bounds into an
optimization problem over three parameters: the truncation parameter
($\alpha$), the smoothing parameter ($\beta$), and an Orlicz parameter
($\rho$). To this end, tools from the theory of N-functions will be employed
to compare N-functions to polynomials.

###  4.1 A modified martingale, and Proof of Theorem 8 ($\mathcal{W}_{3}$
bounds)

The goal of this subsection is to modify the original mds
$\\{Y_{i}\\}_{i=1}^{n}$ to a new mds $\hat{Y}=\\{\hat{Y}_{i}\\}_{i=1}^{n+1}$
which is uniformly bounded except the last term. Throughout this subsection,
we let $\alpha\in(0,\infty]$ be any fixed constant.

Define a martingale difference sequence $Z=Z^{(\alpha)}=\\{Z_{i}\\}_{i=1}^{n}$
as

$Z_{i}=Z_{i}^{(\alpha)}:=Y_{i}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}-E[Y_{i}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}|\mathcal{F}_{i-1}],\quad
1\leq i\leq n.$ (36)

Note that $P(|Z_{i}|\leq\alpha)=1$ for all $1\leq i\leq n$ and, with
$\sigma_{i}^{2}(Z):=E[Z_{i}^{2}|\mathcal{F}_{i-1}]$,

$\displaystyle\sum_{i=1}^{n}\sigma_{i}^{2}(Z)$
$\displaystyle\leq\sum_{i=1}^{n}E[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}]\leq\sum_{i=1}^{n}\sigma_{i}^{2}(Y).$
(37)

Define a stopping time (with the convention $\inf\emptyset=\infty$)

$T=\inf\big{\\{}1\leq m\leq
n:\sum_{i=1}^{m}\sigma_{i}^{2}(Z)>s_{n}^{2}\big{\\}}\wedge(n+1).$ (38)

Since $\sum_{i=1}^{m}\sigma_{i}^{2}(Z)$ is $\mathcal{F}_{m-1}$-measurable,
$(T-1)$ is also a stopping time.

######  Definition 16.

Let $\alpha\in(0,\infty]$ and let $\xi\sim\mathcal{N}(0,1)$ be a standard
normal which is independent of $\\{Y_{i}\\}_{i=1}^{n}$. Define the modified
martingale difference sequence
$\hat{Y}=\hat{Y}^{(\alpha)}=\\{\hat{Y}_{i}\\}_{i=1}^{\infty}$ as follows:
$\forall i\geq 1$,

$\hat{Y}_{i}=\hat{Y}_{i}^{(\alpha)}:=Z_{i}\mathbbm{1}_{\sum_{j=1}^{i}\sigma_{j}^{2}(Z)\leq
1}+\xi\left(s_{n}^{2}-\sum_{j=1}^{T-1}\sigma_{j}^{2}(Z)\right)^{1/2}\mathbbm{1}_{i=n+1}.$
(39)

Enlarging the $\sigma$-fields $\\{\mathcal{F}_{i}\\}$ if necessary,
$\\{\hat{Y}_{m}\\}_{m\geq 0}$ is still a mds. Let

$\hat{X}_{m}=\frac{1}{s_{n}}\sum_{i=1}^{m}\hat{Y}_{i},\quad m\geq 1,\text{
with }\hat{X}_{0}=0.$

Note that $\hat{Y}_{j}=0$ for all $j\in[T,n]\cup(n+1,\infty)$ and recall that
$T\leq n+1$. We write, for $m\geq 1$,
$\sigma_{m}^{2}(\hat{Y}):=E[\hat{Y}_{m}^{2}|\mathcal{F}_{m-1}]$.

The main feature of the mds $\hat{Y}$ is that $|\hat{Y}_{i}|\leq\alpha$ a.s.
for $1\leq i\leq n$, and

$\sum_{i=1}^{n+1}\sigma_{i}^{2}(\hat{Y})=s_{n}^{2}\quad\text{ almost surely.}$
(40)

######  Proposition 17.

For $\alpha\in(0,\infty]$ and a mds $\\{Y_{i}\\}_{i=1}^{n}$, let
$\\{\hat{Y}_{i}\\}$ and $Z=\\{Z_{i}\\}_{i=1}^{n}$ be as in Definition 16. Let
$r\geq 1$ and let $\psi\in[0,\infty)^{[0,\infty)}$ be an N-function with
$\psi\succcurlyeq x^{r\vee 2}$. Recall the definition of
$L_{\psi}=L_{\psi}(Y)$ in (20). Set $\theta_{\psi}:=\lVert Y\rVert_{\psi}$.

  1. (a)

For $\alpha<\infty$, $r\geq 1$,

$\displaystyle\lvert\mathcal{W}_{r}(X_{n},\mathcal{N})-\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\rvert\lesssim_{r}L_{\psi}+\alpha+\frac{n^{1/2}\alpha}{\psi(\alpha
s_{n}/2\theta_{\psi})^{1/(r\vee 2)}}+\lVert V_{n}^{2}-1\rVert_{r/2}^{1/2}.$

  2. (b)

If $\psi$ satisfies that $x\mapsto\psi(\sqrt{x})$ is convex, then for $r\geq
1$,

$\displaystyle\lvert\mathcal{W}_{r}(X_{n},\mathcal{N})-\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\rvert\lesssim_{r}L_{\psi}+\frac{n^{1/2}\alpha}{\psi(\alpha
s_{n}/2\theta_{\psi})^{1/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}+\lVert
V_{n}^{2}-1\rVert_{r/2}^{1/2}.$

Estimate (b) is slightly better than (a) but with a slightly stronger
condition. Statement (b) is more useful for the case $\alpha=\infty$.

######  Lemma 18.

Let $Z=\\{Z_{i}\\}_{i=1}^{n}$ be as in (36). Recall the definition of the
orlicz norm $\lVert\cdot\rVert_{\psi}$ for a sequence in Definition 3. Then,
for any N-function $\psi$,

$\lVert Z\rVert_{\psi}\leq 2\lVert Y\rVert_{\psi}$

###### Proof.

Without loss of generality, assume $\lVert Y\rVert_{\psi}=1$. It suffices to
show that

$E\sum_{i=1}^{n}\psi(|Z_{i}|/2)\leq n.$ (41)

Indeed, recall $E_{i}[\cdot]$ in (27). By the definition of $Z_{i}$ in (36),
for $1\leq i\leq n$,

$\displaystyle E_{i}[\psi(|Z_{i}|/2)]$ $\displaystyle\leq
E_{i}\left[\psi\left((|Y_{i}|\mathbbm{1}_{|Y_{i}|\leq\alpha/2}+E_{i}[|Y_{i}|\mathbbm{1}_{|Y_{i}|\leq\alpha/2}])/2\right)\right]$
$\displaystyle\leq\frac{1}{2}E_{i}\left[\psi(|Y_{i}|\mathbbm{1}_{|Y_{i}|\leq\alpha/2})+\psi(E_{i}[|Y_{i}|\mathbbm{1}_{|Y_{i}|\leq\alpha/2}])\right]$
$\displaystyle\leq
E_{i}\left[\psi(|Y_{i}|\mathbbm{1}_{|Y_{i}|\leq\alpha/2})\right],$

where we used Jensen’s inequality in both the second and the third
inequalities. Summing both sides over all $1\leq i\leq n$, inequality (41)
follows. ∎

###### Proof of Proposition 17.

Throughout, notice that $r\geq 1$ and $\psi\succcurlyeq x^{r\vee 2}$.

Without loss of generality, assume $s_{n}=1$. Set

$\displaystyle K_{r}:=\lVert V_{n}^{2}-1\rVert_{r/2}^{1/2}.$ (42)

To prove Proposition 17, it suffices to show that

$\lVert
X_{n}-\hat{X}_{n+1}\rVert_{r}\lesssim_{r}L_{\psi}+\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\frac{n^{1/2}\alpha}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee
2)}}\mathbbm{1}_{\alpha<\infty}.$ (43)

Indeed, trivially we have
$\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\leq\alpha$. If $\psi(\sqrt{x})$ is
convex, we claim that

$\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\lesssim L_{\psi}.$ (44)

To this end, we apply Lemma A.2 to get

$\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\leq
2^{1/r}\lVert\\{\sigma_{i}(Z)\\}_{i=1}^{n}\rVert_{\psi}\psi^{-1}(n)$

Notice that, when $\psi(\sqrt{x})$ is convex, then using Jensen’s inequality
as in (33), we have
$\lVert\\{\sigma_{i}(Z)\\}_{i=1}^{n}\rVert_{\psi}\leq\lVert Z\rVert_{\psi}$
which is bounded by $2\lVert Y\rVert_{\psi}$ by Lemma 18. Claim (44) follows.

The rest of the proof is devoted to obtain (43).

Recall $T$ in (38). Since $\\{\hat{Y}_{i}\\}$ and $\\{Z_{i}\\}$ coincide up to
$i=T-1$, we have

$\displaystyle X_{n}-\hat{X}_{n+1}$
$\displaystyle=\sum_{i=T}^{n}Z_{i}+\sum_{i=1}^{n}(Y_{i}-Z_{i})-\hat{Y}_{n+1}.$
(45)

Step 1. We will bound the first term $\sum_{i=T}^{n}Z_{i}$ in (45) by

$\lVert\sum_{i=T}^{n}Z_{i}\rVert_{r}\lesssim_{r}K_{r}+L_{\psi},\quad r\geq 1.$
(46)

Since $T-1$ is a stopping time, $\\{Z_{i}\\}_{i=T}^{n}$ form a martingale
difference sequence. Hence, by an inequality of Burkholder Theorem A.3 and
Lemma A.2, for $r\geq 1$,

$\displaystyle\lVert\sum_{i=T}^{n}Z_{i}\rVert_{r}\leq\lVert\sum_{i=T+1}^{n}Z_{i}\rVert_{r}+\lVert\max_{i=1}^{n}|Z_{i}|\rVert_{r}\lesssim_{r}\lVert\sum_{i=T+1}^{n}\sigma_{i}^{2}(Z)\rVert_{r/2}^{1/2}+L_{\psi}.$
(47)

Further, by the definition of the stopping time $T$,

$\displaystyle\sum_{i=T+1}^{n}\sigma_{i}^{2}(Z)=\left(\sum_{i=1}^{n}\sigma_{i}^{2}(Z)-\sum_{i=1}^{T}\sigma_{i}^{2}(Z)\right)\vee
0\leq\left(\sum_{i=1}^{n}\sigma_{i}^{2}(Z)-1\right)\vee 0.$

This inequality, together with (47), yields (46).

Step 2. Consider the second $\sum_{i=1}^{n}(Y_{i}-Z_{i})$ in (45). We will
show that

$\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})\rVert_{r}\lesssim_{r}\frac{\alpha
n^{1/2}}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee
2)}}\mathbbm{1}_{\alpha<\infty}.$ (48)

Indeed, this inequality is trivial for $\alpha=\infty$. When $\alpha<\infty$,
note that $\\{Y_{i}-Z_{i}\\}_{i=1}^{n}$ is a martingale difference sequence.
For $r\geq 1$, by Burkholder’s inequality,

$\displaystyle\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})\rVert_{r}\lesssim_{r}\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}^{1/2}$

Further, by Hölder’s inequality, Jensen’s inequality, and the fact
$Y_{i}-Z_{i}=Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}-E[Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}|\mathcal{F}_{i-1}]$,
we get, for $r\geq 2$,

$\displaystyle\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}\leq
n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}-Z_{i}|^{r}\bigg{\rVert}_{1}^{2/r}\leq
4n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\bigg{\rVert}_{1}^{2/r}.$

Since
$|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\leq\psi(|Y_{i}|/\theta_{\psi})(\alpha/2)^{r}/\psi(\alpha/2\theta_{\psi})$,
we have

$\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\bigg{\rVert}_{1}\leq\frac{(\alpha/2)^{r}}{n\psi(\alpha/2\theta_{\psi})}\sum_{i=1}^{n}E[\psi(|Y_{i}|/2\theta_{\psi})]\leq\frac{(\alpha/2)^{r}}{\psi(\alpha/2\theta_{\psi})}$
(49)

where the last inequality used the definition of $\lVert Y\rVert_{\psi}$.
Hence

$\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}\leq\frac{\alpha^{2}n}{\psi(\alpha/2\theta_{\psi})^{2/(r\vee
2)}}.$

Inequality (48) is proved.

Step 3. Consider the last term in (45). By definition (39),

$\lVert\hat{Y}_{n+1}\rVert_{r}\lesssim_{r}\bigg{\lVert}\big{(}1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\big{)}^{1/2}\bigg{\rVert}_{r}.$

When $T\leq n$, we have

$0\leq 1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\leq
1-\sum_{i=1}^{T}\sigma_{i}^{2}(Z)+\max_{i=1}^{n}\sigma_{i}^{2}(Z)<\max_{i=1}^{n}\sigma_{i}^{2}(Z).$

If $T=n+1$, then

$\Bigr{\lvert}1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\Bigl{\rvert}\leq\Bigr{\lvert}1-\sum_{i=1}^{n}\sigma_{i}^{2}(Y)\Bigl{\rvert}+\Bigr{\lvert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sum_{i=1}^{n}\sigma_{i}^{2}(Z)\Bigl{\rvert}.$

Hence, we have

$\lVert\hat{Y}_{n+1}\rVert_{r}\leq\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}^{1/2}.$
(50)

Step 4. Let us bound the third term in (50). We will show that

$\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}\lesssim\frac{n\alpha^{2}}{\psi(\alpha/2\theta_{\psi})^{2/(r\vee
2)}}\mathbbm{1}_{\alpha<\infty}.$ (51)

This is trivial when $\alpha=\infty$. For $\alpha<\infty$, recall the notation
$E_{i}$ in (27). Then

$\displaystyle\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)=E_{i}[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|>\alpha/2}]+E_{i}[Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}]^{2}\quad\forall
1\leq i\leq n.$

Hence, by Hölder’s inequality and Jensen’s inequality, for $r\geq 2$,

$\displaystyle\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}$
$\displaystyle\leq
2\bigg{\lVert}\sum_{i=1}^{n}E_{i}[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|>\alpha/2}]\bigg{\rVert}_{r/2}$
$\displaystyle\leq
2n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\bigg{\rVert}_{1}^{2/r}\stackrel{{\scriptstyle\eqref{eq:noname}}}{{\lesssim}}\frac{n\alpha^{2}}{\psi(\alpha/2\theta_{\psi})^{2/r}}.$

Display (51) is proved.

Step 5. By (50), (51), we have arrived at

$\lVert\hat{Y}_{n+1}\rVert_{r}\lesssim\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\frac{\alpha
n^{1/2}}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee
2)}}\mathbbm{1}_{\alpha<\infty}.$

This inequality, together with (45), (46), (48), yields (43). ∎

###### Proof of Theorem 8.

Consider $\alpha=\infty$ and recall the definition of
$\hat{Y}=\hat{Y}^{(\infty)}$ in Definition 16. Note that
$Z_{i}^{(\infty)}=Y_{i}$ for $1\leq i\leq n$.

Since $\hat{Y}_{n+1}$ is normal conditioning on $\mathcal{F}_{n}$, by (29) in
Proposition 13 and Remark 14, we have

$\mathcal{Z}_{3}(\hat{X}_{n+1},\mathcal{N})\lesssim\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|\hat{Y}_{i}|^{3}]\leq\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|Z_{i}|^{3}]=\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|Y_{i}|^{3}].$

By (25), this implies $\mathcal{W}_{3}(\hat{X}_{n+1},\mathcal{N})\lesssim
L_{3}$. Further, applying Proposition 17(b) to the case $\alpha=\infty$, we
get

$\mathcal{W}_{3}(X_{n},\mathcal{N})\lesssim\mathcal{W}_{3}(\hat{X}_{n+1},\mathcal{N})+L_{3}+\lVert
V_{n}^{2}-1\rVert_{3/2}^{1/2}.$

Theorem 8 is proved. ∎

###  4.2 Proof of Theorem 5 ($\mathcal{W}_{1}$ bounds)

In what follows we let $\alpha\in(0,\infty),\rho>0$ be constants to be
determined later. Let the mds $\hat{Y}=\hat{Y}^{(\alpha)}$ be as in Definition
16, and set

$\beta=\sqrt{2}\alpha.$ (52)

Throughout this subsection, we simply write

$\displaystyle\sigma_{i}^{2}$
$\displaystyle=\sigma_{i}^{2}(\hat{Y})=E[\hat{Y}_{i}^{2}|\mathcal{F}_{i-1}],\quad
i\geq 1$ $\displaystyle\lambda_{m}^{2}$
$\displaystyle=\sum_{i=m+1}^{n+1}\sigma_{i}^{2}+\beta^{2}s_{n}^{2},\quad 0\leq
m\leq n+1.$ (53)

Note that $\lambda_{0}^{2}=(1+\beta^{2})s_{n}^{2}$, and
$\lambda_{m}^{2}=\lambda_{0}^{2}-\sum_{i=1}^{m}\sigma_{i}^{2}$ is
$\mathcal{F}_{m-1}$-measurable for $m\geq 1$.

Recall that $\hat{Y}_{n+1}$ is normaly distributed given $\mathcal{F}_{i-1}$.
By (40), Proposition 13, and Remark 14, for $r\in\\{1,2\\}$,
$\alpha>0,\beta=\sqrt{2}\alpha$,

$\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha^{r}+s_{n}^{-r}E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}^{3-r}}.$
(54)

The following estimate of the summation within (54) will be crucially employed
in the derivation of our $\mathcal{W}_{r}$ bounds, $r=1,2$.

######  Proposition 19.

Let $\psi\in[0,\infty)^{[0,\infty)}$ be an N-function with $\psi\succcurlyeq
x^{2}$, and set $\theta_{\psi}=\lVert Y\rVert_{\psi}$. Recall
$\alpha,\hat{Y},\lambda_{i}$ above, and $\beta=\sqrt{2}\alpha$. Then, for any
$B,k>0$,

$E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq
B}\lesssim\theta_{\psi}\rho
n+\theta_{\psi}\rho\alpha^{-2}\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge
B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t,\quad\forall\rho>0.$

###### Proof.

Without loss of generality, assume $s_{n}=1$.

Recall that $\lambda_{i}$ is $\mathcal{F}_{i-1}$-measurable. Recall that
$E_{i}[\cdot]=E[\cdot|\mathcal{F}_{i-1}]$ as in (27).

By Young’s inequality (15), for any $\rho>0$,

$E\sum_{i=1}^{n}\frac{E_{i}[|\hat{Y}_{i}|^{3}]}{2\theta_{\psi}\rho\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq
B}\leq
E\sum_{i=1}^{n}E_{i}\big{[}\psi(\tfrac{|\hat{Y}_{i}|}{2\theta_{\psi}})+\psi_{*}(\tfrac{\hat{Y}_{i}^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq
B}\big{]}.$

Note that $\psi_{*}$ is an N-function and so $\psi_{*}\succcurlyeq x$. Using
the fact that $\psi_{*}\succcurlyeq x$ and that $|\hat{Y}_{i}|\leq\alpha$ for
$1\leq i\leq n$, we have, almost surely,

$\psi_{*}(\tfrac{\hat{Y}_{i}^{2}}{\rho\lambda_{i}^{k}})\leq\tfrac{\hat{Y_{i}}^{2}}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}}),\quad\forall
1\leq i\leq n.$

Thus we further have (Note that $\hat{Y}_{i}=0$ for $T\leq i\leq n$.)

$\displaystyle
E\sum_{i=1}^{n}\frac{E_{i}[|\hat{Y}_{i}|^{3}]}{2\theta_{\psi}\rho\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq
B}$ $\displaystyle\leq
E\sum_{i=1}^{n}\psi(\tfrac{|Z_{i}|}{2\theta_{\psi}})+E\sum_{i=1}^{n}\tfrac{E_{i}[\hat{Y}_{i}^{2}]}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq
B}$ $\displaystyle\stackrel{{\scriptstyle
Lemma~{}\ref{lem:orfnorm-z}}}{{\leq}}n+E\sum_{i=1}^{T-1}\tfrac{\sigma_{i}^{2}}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq
B}\quad\forall\rho>0.$

Next, we will show the following integral bound:

$\displaystyle
E\sum_{i=1}^{T-1}\sigma_{i}^{2}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i-1}^{k}})\mathbbm{1}_{\lambda_{i}^{k}\leq
B}\leq\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge
B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t.$ (55)

Recall the sequence of stopping times $\\{\tau_{t}\\}_{0\leq t\leq 1}$ defined
in (35). Then $\tau_{t}=m$ if and only if
$t\in(\sum_{i=1}^{m-1}\sigma_{i}^{2},\sum_{i=1}^{m}\sigma_{i}^{2}]$. Moreover,
recalling that $\beta^{2}=2\alpha^{2}$,

$\lambda_{\tau_{t}}^{2}=1+\beta^{2}-\sum_{i=1}^{\tau_{t}}\sigma_{i}^{2}\geq
1+\beta^{2}-t-\alpha^{2}=1+\alpha^{2}-t.$

Thus, for any $\alpha>0$, writing $v_{m}^{2}:=\sum_{i=1}^{m}\sigma_{i}^{2}$,

$\displaystyle
E\left[\sum_{i=1}^{T-1}\sigma_{i}^{2}\psi_{*}(\tfrac{\alpha^{k}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq
B}\right]$
$\displaystyle=E\sum_{i=1}^{T-1}\int_{v_{i-1}^{2}}^{v_{i}^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{\tau_{t}}^{k}})\mathbbm{1}_{\lambda_{\tau_{t}}^{2}\leq
B}\mathrm{d}t$ $\displaystyle\leq
E\int_{0}^{1}\psi_{*}(\tfrac{\alpha^{2}}{\rho(1+\alpha^{2}-t)^{k/2}})\mathbbm{1}_{1+\alpha^{2}-t\leq
B}\mathrm{d}t$ $\displaystyle\leq\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge
B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t.$

Inequality (55) is proved. The Proposition follows. ∎

To prove Theorem 7 and Theorem 8, we will need the following lemma to compare
N-functions to power functions.

######  Lemma 20.

Let $p>1$ and let $q:=p/(p-1)$ denote its Hölder conjugate. Let $\psi$ be an
N-function. Set
$C_{\psi}=\max_{x,y>0}\frac{\psi^{-1}(x)\psi_{*}^{-1}(x)/x}{\psi^{-1}(y)\psi_{*}^{-1}(y)/y}$.

  1. (1)

If $\psi\preccurlyeq x^{p}$, then

$\frac{\psi_{*}(s)}{s^{q}}\geq
C_{\psi}^{-q}\frac{\psi_{*}(t)}{t^{q}}\quad\text{ for any }s>t>0.$

  2. (2)

If $\psi\succcurlyeq x^{p}$, then

$\frac{\psi_{*}(s)}{s^{q}}\leq
C_{\psi}^{q}\frac{\psi_{*}(t)}{t^{q}}\quad\text{ for any }s>t>0.$

Note that $C_{\psi}=1$ if $\psi=x^{p}$, and (16) guarantees that $1\leq
C_{\psi}\leq 2$ in general.

###### Proof of Theorem 5:.

Without loss of generality, assume $s_{n}=1$. Set $\theta_{\psi}=\lVert
Y\rVert_{\psi}$.

Part 1 (Proof of Theorem 5(i) for general $\psi$).

Since $\psi_{*}(x)/x$ is increasing, we have
$t\psi_{*}(\tfrac{\alpha^{2}}{\rho
t})\leq\alpha^{2}\psi_{*}(\tfrac{\alpha^{2}}{\rho\alpha^{2}})$ for
$t\geq\alpha^{2}$. Hence

$\displaystyle\int_{\alpha^{2}}^{1+\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho
t})\mathrm{d}t\leq\int_{\alpha^{2}}^{1+\alpha^{2}}\frac{\alpha^{2}}{t}\psi_{*}(\tfrac{1}{\rho})\mathrm{d}t=\alpha^{2}\psi_{*}(\tfrac{1}{\rho})\log(1+\tfrac{1}{\alpha^{2}}).$

This inequality, together with Proposition 19, yields

$\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}$
$\displaystyle\lesssim\theta_{\psi}\rho
n+\theta_{\psi}\rho\psi_{*}(\tfrac{1}{\rho})\log(1+\tfrac{1}{\alpha^{2}}).$

Taking $\rho=1\big{/}\psi_{*}^{-1}(n)$ in the above inequality, we obtain

$\displaystyle
E\sum_{i=1}^{T-1}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}\lesssim\theta_{\psi}\rho
n\log(e+\tfrac{1}{\alpha^{2}})\lesssim\theta_{\psi}\psi^{-1}(n)\log(e+\tfrac{1}{\alpha^{2}})$
(56)

where in the last inequality we used the fact

$\rho=1\big{/}\psi_{*}^{-1}(n)\stackrel{{\scriptstyle\eqref{eq:orlicz-
ineq}}}{{\leq}}\tfrac{1}{n}\psi^{-1}(n).$ (57)

Further, putting

$\alpha=2\theta_{\psi}\psi^{-1}(n)=2L_{\psi},$ (58)

then inequality (56) and (54) imply

$\mathcal{W}_{1}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha\log(e+\tfrac{1}{L_{\psi}^{2}}).$

This inequality, together with Proposition 17(a), yields (recall $K_{1}$ in
(42))

$\displaystyle\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\alpha\log(e+\tfrac{1}{L_{\psi}^{2}})+\frac{n^{1/2}\alpha}{\psi(\alpha/2\theta_{\psi})^{1/2}}+K_{1}\stackrel{{\scriptstyle\eqref{eq:alpha-
value}}}{{\lesssim}}\alpha\log(e+\tfrac{1}{L_{\psi}^{2}})+K_{1}.$

Theorem 5(i) is proved.

Part 2 (Proof of Theorem 5(ii) for $\psi\preccurlyeq x^{p}$, $p>2$).

Similar to the previous step, we will first derive a bound for the integral in
(55). The key observation is that we can compare $\psi_{*}$ to $(x^{p})_{*}$
instead of $1/x$ (thanks to Lemma 20).

Let $q:=p/(p-1)$ denote the Hölder conjugate of $p>2$. By Lemma 20,
$\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\leq
C_{\psi}^{q}(\tfrac{\alpha^{2}}{t})^{q}\psi_{*}(\tfrac{\alpha^{2}}{\rho\alpha^{2}})$
for $t>\alpha^{2}$. Hence

$\int_{\alpha^{2}}^{1+\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho
t})\mathrm{d}t\leq\frac{C_{\psi}^{q}}{q-1}\alpha^{2}\psi_{*}(\tfrac{1}{\rho}).$
(59)

This inequality, together with Proposition 19, yields (Note that
$C_{\psi}^{q}\leq 2^{2}$.)

$\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}$
$\displaystyle\lesssim\theta_{\psi}\rho
n+\tfrac{1}{q-1}\theta_{\psi}\rho\psi_{*}(\tfrac{1}{\rho}).$

Putting $\rho=1\big{/}\psi_{*}^{-1}(n)$, this inequality becomes

$\displaystyle
E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}\lesssim
p\theta_{\psi}\rho n\stackrel{{\scriptstyle\eqref{eq:rho-
upper}}}{{\lesssim}}p\theta_{\psi}\psi^{-1}(n).$ (60)

Hence, taking $\alpha=2\theta_{\psi}\psi^{-1}(n)=2L_{\psi}$, inequalities (60)
and (54) imply

$\mathcal{W}_{1}(\hat{X}_{n+1},\mathcal{N})\lesssim p\alpha.$

This inequality, together with Proposition 17(a), yields Theorem 5(ii). ∎

###  4.3 Proof of Theorem 7 ($\mathcal{W}_{2}$ bounds)

In some sense, our proof of the $\mathcal{W}_{2}$ bound in Theorem 7 is an
interpolation between the proofs of Theorem 5 and Theorem 8. We observe that,
to bound $E\sum_{i=1}^{n}|Y_{i}|^{3}/\lambda_{i}$, a better estimate than
Proposition 19 can be achieved by simply bounding $|Y_{i}|^{3}/\lambda_{i}$ by
$|Y_{i}|^{3}/C$ when $\lambda_{i}$ is larger than some “threshold” $C$.

###### Proof of Theorem 7(i):.

Without loss of generality, assume $s_{n}=1$. Set $\theta_{\psi}=\lVert
Y\rVert_{\psi}$.

Define the mds $\hat{Y}=\hat{Y}^{(\infty)}$ and the martingale
$\\{\hat{X}_{i}\\}$ as in Definition 16. Note that $\hat{Y}_{i}=Y_{i}$ for all
$i<T$.

Let $\beta>0$ be a constant to be determined later, and recall the notations
$\sigma_{i},\lambda_{i}$ in (53). Since $\psi\succcurlyeq x^{2}$ and
$\lambda_{i}\geq\beta$ for $1\leq i\leq n+1$, we have
$\lambda_{i}^{2}/\psi(\lambda_{i}/\theta_{\psi})\leq\beta^{2}/\psi(\beta/\theta_{\psi})$
for all $i$. Applying Proposition 13 and Remark 14, we get

$\displaystyle\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})$
$\displaystyle\lesssim\beta^{2}+E\sum_{i=1}^{n}\frac{\lambda_{i}^{2}\psi(|\hat{Y}_{i}|/\theta_{\psi})}{\psi(\lambda_{i}/\theta_{\psi})}$
$\displaystyle\lesssim\beta^{2}+E\sum_{i=1}^{n}\frac{\beta^{2}\psi(|Y_{i}|/\theta_{\psi})}{\psi(\beta/\theta_{\psi})}$
$\displaystyle\lesssim\beta^{2}+\frac{n\beta^{2}}{\psi(\beta/\theta_{\psi})}.$

Taking $\beta=L_{\psi}=\theta_{\psi}\psi^{-1}(n)$, we get

$\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim L_{\psi}^{2}.$

Finally, by Proposition 17(b) and (25), with $\alpha=\infty$,

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})^{1/2}+L_{\psi}+\lVert\sum_{i=1}^{n}\sigma_{i}^{2}-1\rVert_{1}^{1/2}.$

Theorem 7(i) follows. ∎

###### Proof of Theorem 7(ii):.

Without loss of generality, assume $s_{n}=1$.

Let $\alpha>0$ be a constant to be determined. We define the mds
$\hat{Y}=\hat{Y}^{(\alpha)}$ and the martingale $\\{\hat{X}_{i}\\}$ as in
Definition 16. Set $\theta_{\psi}=\lVert Y\rVert_{\psi}$.

Step 1. Applying Proposition 13 and Remark 14 to the case
$\beta=\sqrt{2}\alpha$, we get

$\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha^{2}+E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}.$
(61)

Let $k>0$ be a constant to be determined later, and define and event

$A_{i}=\\{\lambda_{i}\leq k\alpha\\}.$

By Proposition 19,

$E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{\lambda_{i}\leq
k\alpha}\lesssim\theta_{\psi}\rho
n+\theta_{\psi}\rho\alpha^{-2}\int_{0}^{k^{2}\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho
t^{1/2}})\mathrm{d}t$ (62)

Since $\psi\succcurlyeq x^{3}$, by Lemma 20,
$t^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho
t})\lesssim(k\alpha)^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho k\alpha})$ for
$t\in(0,k\alpha)$. Thus

$\displaystyle\int_{0}^{k\alpha}t\psi_{*}(\tfrac{\alpha^{2}}{\rho
t})\mathrm{d}t\lesssim(k\alpha)^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho
k\alpha})\int_{0}^{k\alpha}t^{1-3/2}\mathrm{d}t\lesssim(k\alpha)^{2}\psi_{*}(\tfrac{\alpha}{\rho
k}).$

This inequality, together with (62), yields

$E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}}\lesssim\theta_{\psi}\rho
n+\theta_{\psi}\rho k^{2}\psi_{*}(\tfrac{\alpha}{\rho k}).$

Taking $\rho$ such that $n=4k^{2}\psi_{*}(\tfrac{\alpha}{\rho k})$, i.e.,

$\rho=\frac{\alpha/k}{\psi_{*}^{-1}(n/k^{2})}\stackrel{{\scriptstyle\eqref{eq:orlicz-
ineq}}}{{\leq}}\frac{4\alpha k}{n}\psi^{-1}(\frac{n}{4k^{2}}),$

we obtain

$E\sum_{i=1}^{T-1}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}}\lesssim\theta_{\psi}\rho
n\lesssim\theta_{\psi}\alpha k\psi^{-1}(\frac{n}{4k^{2}}).$ (63)

Step 2. By Lemma 18,
$\lVert\\{\hat{Y}_{i}\\}_{i=1}^{n}\rVert_{3}^{3}\lesssim\lVert
Y\rVert_{3}^{3}$. Recalling that $A_{i}^{c}=\\{\lambda_{i}>k\alpha\\}$ and
$\psi\succcurlyeq x^{3}$,

$\displaystyle
E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}^{c}}\leq
E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{k\alpha}\lesssim\frac{n\lVert
Y\rVert_{3}^{3}}{k\alpha}\lesssim_{\psi(1)}\frac{n\theta_{\psi}^{3}}{k\alpha}.$

where Lemma A.1 is used in the last inequality. This inequality, together with
(63), yields

$E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\lesssim_{\psi(1)}\theta_{\psi}\alpha
k\psi^{-1}(\frac{n}{4k^{2}})+\frac{n\theta_{\psi}^{3}}{k\alpha}.$

Let $g(x):=\frac{1}{x}\psi(x)$. Choosing $k$ such that $\theta_{\psi}\alpha
k\psi^{-1}(\frac{n}{4k^{2}})=\frac{n\theta_{\psi}^{3}}{k\alpha}$, i.e.,

$\frac{1}{\alpha
k}=\sqrt{g^{-1}(\tfrac{\alpha^{2}}{4\theta_{\psi}^{2}})/(n\theta_{\psi}^{2})},$

we arrive at the bound

$E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\lesssim_{\psi(1)}\theta_{\psi}^{2}\sqrt{ng^{-1}(\tfrac{\alpha^{2}}{4\theta_{\psi}^{2}})}.$

Step 3. Further, by (61), we get

$\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim_{\psi(1)}\alpha^{2}+\theta_{\psi}^{2}\sqrt{ng^{-1}(\tfrac{\alpha^{2}}{\theta_{\psi}^{2}})}\,.$

Thus, choosing $\alpha=2L_{\psi}=2\theta_{\psi}\psi^{-1}(n)$, by Proposition
17(a) and (25),

$\displaystyle\mathcal{W}_{2}(X_{n},\mathcal{N})$
$\displaystyle\lesssim\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})^{1/2}+L_{\psi}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2}$
$\displaystyle\lesssim_{\psi(1)}L_{\psi}+\theta_{\psi}\left[ng^{-1}(\psi^{-1}(n)^{2})\right]^{1/4}+\lVert
V_{n}^{2}-1\rVert_{1}^{1/2}.$

Theorem 7(ii) is proved. ∎

##  5 Optimality of the $\mathcal{W}_{1}$ rates: Proof of Proposition 10

For any $p>2$, we will construct a mds $\\{Y_{i}\\}_{i=1}^{n}$ such that
$V_{n}^{2}=1$ and $\mathcal{W}_{1}(X_{n},\mathcal{N})\gtrsim_{p}L_{p}(Y)$.
Note that in our examples, $L_{p}(Y)\to 0$ as $n\to\infty$, justifying the
optimality of the term $L_{\psi}$ in Theorem 5.

We choose

$\alpha=\frac{1}{\log n}.$

(Actually for any $p>2$, any $\alpha\gg n^{1/p-1/2}$ so that $\alpha_{n}\to 0$
as $n\to\infty$ would work.)

######  Example 21.

For $n\geq 3$, let $Y_{1},\ldots,Y_{n-2},Y_{n},\xi,\eta$ be independent random
variables such that

  * •

$Y_{n},\xi\sim\mathcal{N}(0,\alpha_{n}^{2})$,

  * •

$Y_{1},\ldots,Y_{n-2}$ are i.i.d.
$\mathcal{N}(0,\tfrac{1-2\alpha_{n}^{2}}{n-2})$ normal random variables,

  * •

$\eta$ has distribution

$P(\eta=\tfrac{1}{2})=\tfrac{4}{5},\quad P(\eta=-2)=\tfrac{1}{5},$

We let $X_{m}:=Y_{1}+\ldots+Y_{m}$ and define a set $A\subset\mathbb{R}$ as

$A=\left\\{x\in\mathbb{R}:\cos
x\geq\tfrac{1}{2}\right\\}=\bigcup_{k\in\mathbb{Z}}\big{[}(2k-\tfrac{1}{3})\pi,(2k+\tfrac{1}{3})\pi\big{]}.$

Define

$Y_{n-1}=\xi\mathbbm{1}_{X_{n-2}\notin\alpha
A}+\alpha\eta\mathbbm{1}_{X_{n-2}\in\alpha A}.$

Clearly, $\\{X_{m}\\}_{m=1}^{n}$ is a martingale, and

$\sigma_{n-1}^{2}=\sigma_{n}^{2}=\alpha^{2},\quad\sigma_{i}^{2}=\tfrac{1-2\alpha^{2}}{n-2}\,\text{
for }i=1,\ldots,n-2.$

Of course, $V_{n}^{2}=\sum_{i=1}^{n}\sigma_{i}^{2}=1$ almost surely, and

$L_{p}=\left(\sum_{i=1}^{n}E[|Y_{i}|^{p}]\right)^{1/p}\leq\left((n-2)\frac{c}{n^{p/2}}+c\alpha^{p}\right)^{1/p}\lesssim_{p}\alpha.$

###### Proof of Proposition 10(1).

Let $\\{Y_{i}\\}_{i=1}^{n}$ be the mds in Example 21. We consider the function

$h(x)=\alpha\sin(\tfrac{x}{\alpha}).$

Clearly, $[h]_{0,1}\leq 1$. Recall the definition of the function
$h_{\alpha}(x)$ in (26). Then

$h_{\alpha}(x)=\alpha\int_{\mathbb{R}}\sin\big{(}\tfrac{1}{\alpha}(x-\alpha
u)\big{)}\phi(u)\mathrm{d}u=h(x)\int_{\mathbb{R}}\cos
u\phi(u)\mathrm{d}u=\frac{1}{\sqrt{e}}h(x),$ (64)

where $\phi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ denotes the standard normal
density. Moreover,

$\displaystyle
E[h(X_{n})-h(\mathcal{N})]=E[h_{\alpha}(X_{n-1})-h_{\alpha}(X_{n-2}+\alpha\mathcal{N})].$

By the definition of $Y_{n-1}$, we have

$\displaystyle
E[h_{\alpha}(X_{n-1})]=E[h_{\alpha}(X_{n-2}+\alpha\mathcal{N})\mathbbm{1}_{X_{n-2}\notin\alpha
A}]+E[h_{\alpha}(X_{n-1}+\alpha\eta)\mathbbm{1}_{X_{n-2}\in\alpha A}].$

Thus

$\displaystyle E[h(X_{n})-h(\mathcal{N})]$ (65)
$\displaystyle=E\left[\left(h_{\alpha}(X_{n-2}+\alpha\eta)-h_{\alpha}(X_{n-2}+\alpha\mathcal{N})\right)\mathbbm{1}_{X_{n-1}\in\alpha
A}\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:calcul-h}}}{{=}}\tfrac{\alpha}{\sqrt{e}}E\left[\left(\sin\tfrac{X_{n-2}}{\alpha}(\cos\eta-\cos\mathcal{N})+\cos\tfrac{X_{n-2}}{\alpha}(\sin\eta-\sin\mathcal{N})\right)\mathbbm{1}_{X_{n-2}\in\alpha
A}\right]$
$\displaystyle=\tfrac{\alpha}{\sqrt{e}}\left(E\left[\sin\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha
A}\right]E[\cos\eta-\cos\mathcal{N}]+E\left[\cos\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha
A}\right]E[\sin\eta]\right).$

Since the set $A$ is symmetric, i.e., $A=-A$, and
$X_{n-2}\sim\mathcal{N}(0,1-2\alpha^{2})$, we get
$E\left[\sin\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha A}\right]=0$.
Hence

$\displaystyle E[h(X_{n})-h(\mathcal{N})]$
$\displaystyle=\tfrac{\alpha}{\sqrt{e}}E\left[\cos\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha
A}\right]E[\sin\eta]$
$\displaystyle\geq\tfrac{\alpha}{2\sqrt{e}}P(X_{n-2}\in\alpha A)E[\sin\eta].$

Note that $E[\sin\eta]=\tfrac{4}{5}\sin\tfrac{1}{2}-\tfrac{1}{5}\sin 2>0.2$,
and, writing $c_{1}:=(1-2\alpha^{2})^{-1/2}$,

$\displaystyle P(X_{n-2}\in\alpha A)$ $\displaystyle=P(\mathcal{N}\in
c_{1}\alpha A)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\int_{c_{1}(6k-1)\alpha\pi/3}^{c_{1}(6k+1)\alpha\pi/3}\phi(u)\mathrm{d}u$
$\displaystyle\geq\frac{1}{2}\left(\int_{0}^{c_{1}\alpha\pi/3}\phi(u)\mathrm{d}u+\sum_{k=0}^{\infty}\tfrac{2c_{1}\alpha\pi}{3}\phi(\tfrac{c_{1}(6k+1)\alpha\pi}{3})\right)$
$\displaystyle\geq\frac{1}{6}\int_{0}^{\infty}\phi(u)\mathrm{d}u=\frac{1}{12}.$
(66)

Therefore, $E[h(X_{n})-h(\mathcal{N})]\geq\frac{\alpha}{120\sqrt{e}}$ and so

$\mathcal{W}_{1}(X_{n},\mathcal{N})\geq\frac{\alpha}{120\sqrt{e}}\geq CL_{p},$

justifying the optimality of the power of $L_{\psi}$ in Theorem 5. ∎

We can modify Example 21 above to justify the optimality of the exponent of
$\lVert V^{2}-1\rVert_{1/2}$ in Theorem 5.

######  Example 22.

For $n\geq 3$, let $Y_{1},\ldots,Y_{n-2}$ be i.i.d. with
$Y_{1}\sim\mathcal{N}(0,\frac{1-\alpha^{2}}{n-2})$. Let

$Y_{n-1}=\alpha\eta\mathbbm{1}_{X_{n-2}\in\alpha A},$

where $X_{n-2}=\sum_{i=1}^{n-1}Y_{i}$, $\alpha=\alpha_{n}=1/\log n$ , $\eta$
and the set $A\subset\mathbb{R}$ are as in Example 21. Let
$Y_{n}\sim\mathcal{N}\left(0,\alpha^{2}P(\mathcal{N}\notin\kappa A)\right)$,
where $\kappa=\kappa_{n}=\alpha/\sqrt{1-\alpha^{2}}$.

###### Proof of Proposition 10(2).

Let the mds $\\{Y_{i}\\}_{i=1}^{n}$ be as in Example 22.

Clearly,
$L_{3}=\left(\sum_{i=1}^{n}E[|Y_{i}|^{3}]\right)^{1/3}\lesssim\alpha$, and

$\displaystyle\lVert V_{n}^{2}-1\rVert_{1/2}$
$\displaystyle=\alpha^{2}E\left[|\mathbbm{1}_{X_{n-2}\in\alpha
A}-P(X_{n-2}\in\alpha A)|^{1/2}\right]^{2}$
$\displaystyle\asymp\alpha^{2}P(\mathcal{N}\notin\kappa
A)P(\mathcal{N}\in\kappa A).$

Hence, by Theorem 5(ii), we get

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\alpha,$ (67)

Next we will show that, there exists $N>0$ such that for $n>N$,

$\lVert V_{n}^{2}-1\rVert_{1/2}\asymp\alpha^{2}.$ (68)

Note that the same computation as in (5) yields $P(\mathcal{N}\in\kappa A)\geq
1/12$. Hence we only need to show that $P(\mathcal{N}\notin\kappa A)\gtrsim
1$. Indeed,

$\displaystyle P(\mathcal{N}\in\kappa A)$
$\displaystyle=\sum_{i\in\mathbb{Z}}\int_{c_{1}(6i-1)\kappa\pi/3}^{c_{1}(6i+1)\kappa\pi/3}\phi(u)\mathrm{d}u$
$\displaystyle\leq
2\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+2\sum_{i=1}^{\infty}\frac{2c_{1}\kappa\pi}{3}\phi(\frac{c_{1}(6i-1)\kappa\pi}{3})$
$\displaystyle\leq
2\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+\frac{2}{3}\int_{-c_{1}\kappa\pi/3}^{\infty}\phi(u)\mathrm{d}u$
$\displaystyle\leq\frac{8}{3}\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+\frac{1}{3}\leq\frac{1}{2}$

for all $n$ sufficiently big. Inequality (68) follows.

It remains to show that

$\mathcal{W}_{1}(X_{n},\mathcal{N})\gtrsim\alpha.$

To this end, we consider the function $h(x)=\alpha\sin(\tfrac{x}{\alpha})$.
Define $\lambda>0$ by

$\lambda^{2}={\rm Var}(Y_{n})=\alpha^{2}P(\mathcal{N}\notin\kappa
A)\in[\tfrac{1}{2}\alpha^{2},\tfrac{11}{12}\alpha^{2}].$

Recall the definition of $h_{\lambda}(x)$ in (26). By the same calculation as
in (64),

$h_{\lambda}(x)=h(x)\int_{\mathbb{R}}\cos(\tfrac{\lambda}{\alpha}u)\phi(u)\mathrm{d}u=h(x)\exp(-\frac{\lambda^{2}}{2\alpha^{2}}).$

By symmetry, $E[h(\mathcal{N})]=0$ and
$E[h_{\lambda}(X_{n-2})\mathbbm{1}_{X_{n-2}\notin\alpha A}]=0$. Hence

$\displaystyle E[h(X_{n})-h(\mathcal{N})]$
$\displaystyle=E[h_{\lambda}(X_{n-2})\mathbbm{1}_{X_{n-2}\notin\alpha
A}]+E[h_{\lambda}(X_{n-2}+\alpha\eta)\mathbbm{1}_{X_{n-2}\in\alpha A}]$
$\displaystyle=\alpha\exp(-\frac{\lambda^{2}}{2\alpha^{2}})E[\cos(\tfrac{X_{n-2}}{\alpha})\sin\eta\mathbbm{1}_{X_{n-2}\in\alpha
A}]$ $\displaystyle\geq
0.1\alpha\exp(-\frac{\lambda^{2}}{2\alpha^{2}})P(X_{n-2}\in\alpha A)\geq
c\alpha.$

Display (67) is proved. Our proof of the Proposition is complete. ∎

##  6 Some open questions

  1. 1.

For the $\mathscr{L}^{\infty}$ case (i.e. the mds is uniformly
$\mathscr{L}^{\infty}$-bounded), is the typical $\mathcal{W}_{1}$ rates
$O(n^{-1/2}\log n)$ within Theorem 5 optimal? Recall that it is shown by
Bolthausen [5] that this rate is typically optimal for the Kolmogorov
distance.

  2. 2.

For the $\mathscr{L}^{\infty}$ case, are the typical $\mathcal{W}_{r}$ rates
$O(n^{-/(2r)})$, $r\in\\{2,3\\}$, in Theorems 7 and 8 optimal? For general
$r>3$ and bounded mds, can we get similar bounds $O(n^{-/(2r)})$, $r>3$?

  3. 3.

Can we say anything about the optimality of the $\mathcal{W}_{2}$ rates in
Theorem 7 when the mds is $\mathscr{L}^{p}$ integrable, for any $p>2$?

  4. 4.

Is there a better (unified) formula than the “piecewise” $\mathcal{W}_{2}$
bound in Theorem 7 for $\mathscr{L}^{p}$ martingales, $p>2$?

  5. 5.

As a feature of the $\mathcal{W}_{1},\mathcal{W}_{2}$ bounds in Theorems 5 and
7, better integrability implies faster (typical) convergence rates for the
martingale CLT. But this is no longer the case for the $\mathcal{W}_{3}$
estimate in Theorem 8. Is it possible to get better $\mathcal{W}_{3}$ bounds
for general martingales with better integrability than $\mathscr{L}^{3}$?

  6. 6.

How to obtain the Wasserstein rates for the CLT of multi-dimensional
martingales? Can we say something on the dependence of the rates on the
dimension?

  7. 7.

How to obtain the Wasserstein-$r$ convergence rates, $r>3$, for the martingale
CLT in terms of $L_{\psi}$?

##  Appendix A Appendix

###  A.1 Comparison between Orlicz norms

Recall the definition of the (mean-)orlicz norm $\lVert\cdot\rVert_{\psi}$ for
a sequence in Definition 3.

######  Lemma A.1.

Let $p\geq 1$ and let $Y=\\{Y_{i}\\}_{i=1}^{n}$ be a sequence of random
variables with $\lVert Y\rVert_{p}<\infty$. If an N-function $\psi$ satisfies
$\psi\succcurlyeq x^{p}$, then

$\lVert Y\rVert_{p}\leq\left(1+\frac{1}{\psi(1)}\right)^{1/p}\lVert
Y\rVert_{\psi}.$

###### Proof.

Without loss of generality assume $\lVert Y\rVert_{\psi}=1$. Since
$\psi/x^{p}$ is increasing,

$\displaystyle E[|Y_{i}|^{p}]\leq 1+E[|Y_{i}|^{p}\mathbbm{1}_{|Y_{i}|>1}]\leq
1+\frac{1}{\psi(1)}E\left[\psi(|Y_{i}|)\right].$

Thus, by the definition of $\lVert\cdot\rVert_{\psi}$,

$\displaystyle\lVert
Y\rVert_{p}^{p}=\frac{1}{n}\sum_{i=1}^{n}E[|Y_{i}|^{p}]\leq
1+\frac{1}{\psi(1)n}\sum_{i=1}^{n}E[\psi(|Y_{i}|)]\leq 1+\frac{1}{\psi(1)}.$

The lemma follows. ∎

###  A.2 Regularity of Gaussian smoothing:Proof of Lemma 12

The proof is exactly as in [10, Lemma 6.1].

###### Proof.

Since $f_{\sigma}(x)=\int_{\mathbb{R}}f(x-\sigma u)\phi(u)$, integration by
parts yields

$\displaystyle f_{\sigma}^{(k)}(x)$
$\displaystyle=\frac{1}{\sigma^{k}}\int_{\mathbb{R}}f^{(\ell)}(x-\sigma
u)\phi^{(k-\ell)}(u)\mathrm{d}u$
$\displaystyle=\frac{1}{\sigma^{k-\ell}}\int_{\mathbb{R}}[f^{(\ell)}(x-\sigma
u)-f^{(\ell)}(x)]\phi^{(k-\ell)}(u)\mathrm{d}u,\quad\forall 0\leq\ell<k.$

The lemma follows by taking $\ell=r-1$ and using the fact $[f]_{r-1,1}=1$. ∎

###  A.3 Moment bound of the maximum

######  Lemma A.2.

Let $\psi\succcurlyeq x^{r}$ be an N-function, $r>0$. For any sequence of
random variables $Y=\\{Y_{i}\\}_{i=1}^{n}$, we have

$\lVert\max_{i=1}^{n}|Y_{i}|\rVert_{r}\leq 2^{1/r}\lVert
Y\rVert_{\psi}\psi^{-1}(n).$

###### Proof.

Without loss of generality, assume $\lVert Y\rVert_{\psi}=1$. For any
$\alpha>0$,

$|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|\geq\alpha}\leq\frac{\alpha^{r}}{\psi(\alpha)}\psi(|Y_{i}|)\mathbbm{1}_{|Y_{i}|\geq\alpha},\quad
1\leq i\leq n$

where we used the fact that $x\mapsto\frac{\psi}{x^{r}}$ is increasing. Hence

$\displaystyle E[\max_{i=1}^{n}|Y_{i}|^{r}]$
$\displaystyle\leq\alpha^{r}+E[\max_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|\geq\alpha}]$
$\displaystyle\leq\alpha^{r}+E\left[\sum_{i=1}^{n}\frac{\alpha^{r}}{\psi(\alpha)}\psi(|Y_{i}|)\right]\leq\alpha^{r}+\frac{n\alpha^{r}}{\psi(\alpha)}$

Taking $\alpha=\psi^{-1}(n)$, we get
$\lVert\max_{i=1}^{n}|Y_{i}|\rVert_{r}^{r}\leq 2\alpha^{r}$. ∎

###  A.4 Proof of Lemma 20

###### Proof.

We only give the proof of (1). The proof of (2) is similar.

The case $\psi=x^{p}$ is trivial, so we only consider the general case
$\psi\preccurlyeq x^{p}$. It suffices to show that, for any $s>t>0$,

$\frac{s^{1/q}}{\psi_{*}^{-1}(s)}\geq\frac{1}{4}\frac{t^{1/q}}{\psi_{*}^{-1}(t)},$

which, writing $F(x):=\psi^{-1}(x)\psi_{*}^{-1}(x)$, is equivalent to

$\frac{\psi^{-1}(s)/s^{1/p}}{\psi^{-1}(t)/t^{1/p}}\geq\frac{1}{C_{\psi}}\frac{F(s)/s}{F(t)/t}\quad\forall
s>t>0.$

Note that $\psi\preccurlyeq x^{p}$ implies that the left-side is at least 1.
Lemma 20(1) follows. ∎

###  A.5 A Burkholder inequality

######  Theorem A.3.

Let $\\{Y_{i}\\}_{i=1}^{n}$ be a mds and let $S_{m}=Y_{1}+\ldots+Y_{m}$,
$1\leq m\leq n$. Recall the notation $\sigma_{i}(Y)^{2}$ in (1). Then, fir
$p>0$,

$E[\max_{i=1}^{n}|S_{n}|^{p}]\lesssim_{p}E[\big{(}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)\big{)}^{p/2}]+E[\max_{i=1}^{n}|Y_{i}|^{p}].$

This version is taken from [27, Theorem 2.11]. For a more general inequality,
see [7, Theorem 21.1].

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