# Nonuniform Berry-Esseen bounds for Studentized U-statistics

Dennis Leung  School of Mathematics and Statistics, University of Melbourne
<EMAIL_ADDRESS>and  Qi-Man Shao  Department of Statistics and
Data Science, SICM, National Center for Applied Mathematics Shenzhen, Southern
University of Science and Technology<EMAIL_ADDRESS>
###### Abstract.

We establish _nonuniform_ Berry-Esseen (B-E) bounds for Studentized
U-statistics of the rate $1/\sqrt{n}$ under a third-moment assumption, which
covers the t-statistic that corresponds to a kernel of degree $1$ as a special
case. While an interesting data example raised by Novak (2005) can show that
the form of the nonuniform bound for standardized U-statistics is actually
_invalid_ for their Studentized counterparts, our main results suggest that,
the validity of such a bound can be restored by minimally augmenting it with
an additive correction term that decays exponentially in $n$. To our best
knowledge, this is the first time that valid nonuniform B-E bounds for
Studentized U-statistics have appeared in the literature.

###### Key words and phrases:

Exponential lower tail bound of non-negative kernel U-statistics, nonlinear
statistics, nonuniform Berry-Esseen bound, Stein’s method, Studentization,
U-statistics, variable censoring

###### 2000 Mathematics Subject Classification:

62E17

##  1\. Introduction

Let $X_{1},\dots,X_{n}\in\operatorname{\mathcal{X}}$ be independent and
identically distributed (i.i.d.) random variables taking values in a
measurable space
$(\operatorname{\mathcal{X}},\Sigma_{\operatorname{\mathcal{X}}})$. A
U-statistic (Hoeffding, 1948) of degree $m\geq 1$ is defined as

$U_{n}={n\choose m}^{-1}\sum_{1\leq i_{1}<\cdots<i_{m}\leq
n}h(X_{i_{1}},\dots,X_{i_{m}}),$

where $h:\operatorname{\mathcal{X}}^{m}\rightarrow\operatorname{\mathbb{R}}$
is a symmetric and measurable function in $m$ arguments, also known as a
_kernel_ function. This important construction covers a wide range of
statistics, including the sample mean $n^{-1}\sum_{i=1}^{n}X_{i}$ as the
simplest example with $m=1$, for which

(1.1) $h(x)=x\text{ and }\operatorname{\mathcal{X}}=\mathbb{R}.$

For the theorems stated in this article, we will throughout assume, without
loss of generality, that

(1.2) $\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})]=0,$

though knowing that such re-centering may not be done in practice because the
mean of $h(\cdot)$ could be unknown. In the U-statistic literature, it is well
established that under the finite second-moment assumption
$\operatorname{\mathbb{E}}[h^{2}(X_{1},\dots,X_{m})]<\infty$ and the _non-
degeneracy condition_

(1.3) $\sigma^{2}\equiv\operatorname{Var}[g(X_{1})]>0,$

where $g(\cdot)$ the first-order _canonical function_ defined by

$g(x)=\operatorname{\mathbb{E}}[h(X_{1},X_{2},\dots,X_{m})|X_{1}=x],$

one has the weak convergence

(1.4) $\frac{\sqrt{n}}{m\sigma}U_{n}\longrightarrow_{d}N(0,1)\text{ as
}n\longrightarrow\infty,$

which extends the classical central limit theorem for the sample mean.

There has always been great interest in characterizing the normal
approximation accuracy of (1.4) by Berry-Esseen (B-E) bounds; see Filippova
(1962), Grams and Serfling (1973), Bickel (1974), Callaert and Janssen (1978),
Chan and Wierman (1977), van Zwet (1984), Friedrich (1989), Chen and Shao
(2007) and Bentkus et al. (1994) for an inexhaustive list of such works. For
instance, Chen and Shao (2007)’s results suggest that, under (1.2), (1.3) and
$\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{3}]<\infty$, when $2m<n$,
one has the bounds

(1.5)
$\sup_{x\in\operatorname{\mathbb{R}}}\left|P\left(\frac{\sqrt{n}}{m\sigma}U_{n}-x\right)-\Phi(x)\right|\leq
C_{1}(m)\frac{\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{3}]}{\sqrt{n}\sigma^{3}}$

and

(1.6) $\left|P\left(\frac{\sqrt{n}}{m\sigma}U_{n}\leq
x\right)-\Phi(x)\right|\leq
C_{2}(m)\frac{\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{3}]}{(1+|x|)^{3}\sqrt{n}\sigma^{3}}\text{
for any }x\in\operatorname{\mathbb{R}},$

where $\Phi(x)$ is the standard normal distribution function, and $C_{1}(m)$
and $C_{2}(m)$ are positive constants depending only on $m$111The moment
quantities in (1.5) and (1.6) have been simplified here for brevity; refer to
Chen and Shao (2007, Section 3.1) for more sophisticated versions of such
bounds.. In contrast to the _uniform_ bound in (1.5), (1.6) is known as a
_nonuniform_ B-E bound, which is qualitatively more informative by having a
”nonuniform” multiplicative factor that decays in the magnitude of $x$.
Without doubt, the sample mean from (1.1) has the richest literature since the
works of Berry (1941) and Esseen (1942), where even the absolute constant’s
value is very well understood (Esseen, 1956, Shevtsova, 2011).

Nevertheless, with some exceptions such as the rank-based Kendall’s tau
statistic (Kendall, 1938) for testing independence and Wilcoxon signed rank
statistic (Wilcoxon, 1945) for testing medians, whose respective degree-two
kernels have $\sigma=1/3$ and $\sigma=1/12$ under a point null conditions like
(1.2) and other regularity assumptions, $\sigma$ is typically unknown and
cannot be directly used to standardize $U_{n}$. It is hence more relevant to
develop a B-E bound for U-statistics that are _Studentized_ with the data-
driven Jackknife estimator of $\sigma$ proposed by Arvesen (1969); in
particular, for the special degree-one kernel in (1.1), the resulting
Studentized U-statistic is precisely the t-statistic of Gosset (Student,
1908). Other typical examples of U-statistics that must require Studentization
are the sample variance and Gini’s mean difference; see Lai et al. (2011,
Section 1) for the forms of their degree-two kernels. The quest for developing
B-E bounds for such Studentized U-statistics has not gone unnoticed by
researchers: Uniform B-E bounds of rate $1/\sqrt{n}$ analogous to (1.5) have
been developed for Studentized U-statistics of degree $2$ by Helmers (1985),
Callaert and Veraverbeke (1981), Zhao (1983) and Wang et al. (2000),
respectively under $4.5$, $4+\varepsilon$ for any $\varepsilon>0$, $4$ and $3$
finite absolute moments imposed on the kernel $h(X_{1},X_{2})$. Most recently,
under $3$ finite absolute moments, we have obtained a uniform B-E bound for
Studentized U-statistics of any degree $m$, and also advocated _variable
censoring_ as the appropriate technical device to prove such bounds under the
Stein-method approach (Leung and Shao, 2023).

To our best knowledge, a nonuniform bound for Studentized U-statistics that is
valid for all $x\in\operatorname{\mathbb{R}}$ in the same spirit as (1.6) is
still eluding the literature, even for the t-statistic and the even simpler
_self-normalized sum_ $S_{n}/V_{n}$, where

(1.7) $S_{n}\equiv\sum_{i=1}^{n}X_{i}\text{ and
}V_{n}^{2}\equiv\sum_{i=1}^{n}X_{i}^{2}\text{ for i.i.d.
}X_{1},\dots,X_{n}\in\operatorname{\mathbb{R}};$

see (2.14) below for a classical algebraic relationship between the
t-statistic and the self-normalized sum. In fact, an earlier nonuniform B-E
bound for the self-normalized sum stated in Wang and Jing (1999, Corollary
2.3) has been latter disproved by an interesting binary data example raised by
Novak (2005, p.342-343), which also demonstrates it is in fact _impossible_ to
have a nonuniform B-E bound of the ”usual form”,

(1.8) $\left|P\Bigg{(}\frac{S_{n}}{V_{n}}\leq
x\Bigg{)}-\Phi(x)\right|\leq\frac{C\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[|X_{1}|^{2}])^{3/2}}d(|x|)\text{
for all }x\in\operatorname{\mathbb{R}},$

that holds for an absolute constant $C$ and any non-increasing function
$d:\operatorname{\mathbb{R}}_{\geq
0}\rightarrow\operatorname{\mathbb{R}}_{\geq 0}$ with the property
$\lim_{x\rightarrow\infty}d(x)=0$, assuming
$\operatorname{\mathbb{E}}[X_{1}]=0$.

This void is now filled by the new nonuniform B-E bound for _Studentized
U-statistics_ of any degree $m$ established in this paper. As we point out in
Section 2, Novak (2005)’s example also readily implies that, for a Studentized
U-statistic $T_{n}$, it is similarly impossible to have a bound of the form:

(1.9) $|P(T_{n}\leq
x)-\Phi(x)|\leq\frac{C(m)\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{3}]}{\sqrt{n}\sigma^{3}}d(|x|)$

that holds universally for all types of data distributions and kernels, where
$C(m)$ is a positive constant depending only on $m$ and $d$ is any non-
increasing function with the same property as the one alluded to in (1.8). As
such, our new nonuniform B-E bound for $T_{n}$ has to give up the form in
(1.9), but, interestingly, not too much; our main theorem (Theorem 3.1)
suggests that, to restore the validity, it suffices to minimally augment the
bound with an additive correction term

$\exp\Bigg{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{3}])^{2}}\Bigg{)}$

that decays exponentially in $n$, for a small constant $c(m)>0$.

Our proof follows Stein’s method, in a similar vein as our work (Leung and
Shao, 2023) on developing uniform B-E bounds for self-normalized nonlinear
statistics. We comment on two major departures in terms of techniques: First,
to elicit the nonuniformity in $x$, considerably more delicate censoring
techniques than the ones in Leung and Shao (2023) have to be employed.
Secondly, to obtain the correction term that decays exponentially in $n$, we
analyze the Jackknife estimate of $\sigma$ by proving an exponential lower-
tail bound developed for U-statistics with non-negative kernels (Lemma 4.3);
the latter result is a crucial technical tool, which naturally extends a
similar result for a sum of non-negative random variables and is of
independent interest.

Organization. Section 2 covers the basics of Studentized U-statistics, and
revisits Novak (2005)’s data example to deduce that the nonuniform bound in
the usual form of (1.9) cannot be valid. Section 3 states our new nonuniform
B-E bounds, including a general one for Studentized U-statistics and a further
refined one for the t-statistic. Sections 4 and 5 respectively prove the two
theorems in Section 3, with the appendices covering additional technical
proofs integral to them.

Notation. For any $p\geq 1$, we use $\|X\|_{p}=(\mathbb{E}|X|^{p})^{1/p}$ to
denote the $L_{p}$-norm of any real-valued random variable $X$; if
$f:\operatorname{\mathcal{X}}^{L}\rightarrow\operatorname{\mathbb{R}}$ is any
function in $L\in\\{1,\dots,n\\}$ arguments, we may use
$\operatorname{\mathbb{E}}[f]$ as shorthand for
$\operatorname{\mathbb{E}}[f(X_{1},\dots,X_{L})]$; likewise, we may use
$\|f\|_{p}$ as a shorthand for the $p$-norm $\|f(X_{1},\dots,X_{L})\|_{p}$. If
$a,b\in\operatorname{\mathbb{R}}$, we let $a\vee b=\max(a,b)$ and $a\wedge
b=\min(a,b)$. $\bar{\Phi}(\cdot)\equiv 1-\Phi(\cdot)$ is the standard normal
survival function, $\phi(\cdot)$ denotes the standard normal density, and
$I(\cdot)$ denotes the indicator function. For any subset
$\operatorname{\mathcal{S}}\subset\\{1,\dots,n\\}$, we shall let
$X_{\operatorname{\mathcal{S}}}\equiv(X_{s})_{s\in\operatorname{\mathcal{S}}}$
be a vector of variables from $X_{1},\dots,X_{n}$ with sample indices in
$\operatorname{\mathcal{S}}$, and
$x_{\operatorname{\mathcal{S}}}=(x_{s})_{s\in\operatorname{\mathcal{S}}}$ be a
similar sub-vector of any generic vector
$(x_{1},\dots,x_{n})\in\operatorname{\mathbb{R}}^{n}$.
$C,c,C_{1},c_{1},C_{2},c_{2}\dots$ denote unspecified _absolute_ positive
constants, where ”absolute” means they are universal for all underlying
distributions of the variables involved and do not depend on other quantities;
if a positive constant does depend on other quantities such as $a$ and/or $b$
_exclusively_ , it will be explicitly specified as $C(a)$, $C(a,b)$, $c(a)$,
$c(a,b)$, etc. to emphasize the dependence on $a$, $(a,b)$, etc. _All these
absolute constants generally differ in values at different occurences_.

##  2\. Studentized U-statistics and Novak (2005)’s example

We first review the basics of Studentized U-statistics. With

$q_{i}=\frac{1}{{n-1\choose m-1}}\sum_{\begin{subarray}{c}1\leq
i_{1}<\dots<i_{m-1}\leq n\\\ i_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}h(X_{i},X_{i_{1}},\dots,X_{i_{m-1}}),\qquad
i=1,\dots,n,$

serving as proxies for the unknown quantities $g(X_{1}),\dots,g(X_{n})$, the
”leave-one-out” Jackknife estimator (Arvesen, 1969) for $\sigma^{2}$ is
constructed as

(2.1)
$\hat{\sigma}^{2}=\frac{n-1}{(n-m)^{2}}\sum_{i=1}^{n}(q_{i}-U_{n})^{2}=\frac{n-1}{(n-m)^{2}}\Big{(}\sum_{i=1}^{n}q_{i}^{2}-nU_{n}^{2}\Big{)}$

to define the _Studentized U-statistic_

(2.2) $\quad T_{n}=\frac{\sqrt{n}}{m\hat{\sigma}}U_{n}.$

For the special case of $m=1$ and the kernel in (1.1), one can check that
$T_{n}$ is precisely the Student’s t-statistic (Student, 1908)

$T_{student}\equiv\frac{\sqrt{n}\bar{X}_{n}}{s_{n}},$

where $\bar{X}_{n}=n^{-1}\sum_{i=1}^{n}X_{i}$ and
$s_{n}^{2}=(n-1)^{-1}\sum_{i=1}^{n}(X_{i}-\bar{X}_{n})^{2}$. It is instructive
to clarify the value taken upon by $T_{n}$ when $\hat{\sigma}$ is equal to
zero, which could be the case for some realizations of the data. The following
convention is adopted:

######  Convention 1 (Convention for $T_{n}$ when $\hat{\sigma}=0$).

  1. (i)

If $\hat{\sigma}=0$ and $U_{n}\neq 0$, $T_{n}$ is assigned the value $+\infty$
or $-\infty$ following the sign of $U_{n}$.

  2. (ii)

If $\hat{\sigma}=0$ and $U_{n}=0$, $T_{n}$ is assigned the value $0$.

Under this convention, there is no ambiguity in understanding an event like
$\\{T_{n}\leq x\\}$ for any $x\in\operatorname{\mathbb{R}}$ and its
probability. Recently, the following _uniform_ B-E bound has been established
for $T_{n}$:

######  Theorem 2.1 (Uniform B-E bound for Studentized U-statistics, Leung and
Shao (2023)).

Assume (1.2)-(1.3), $2m<n$ and $\operatorname{\mathbb{E}}[|h|^{3}]<\infty$.
For a positive absolute constant $C(m)>0$ depending on $m$ only, the following
Berry-Esseen bound holds:

$\sup_{x\in\operatorname{\mathbb{R}}}|P(T_{n}\leq
x)-\Phi(x)|\leq\frac{C(m)}{\sqrt{n}}\left\\{\frac{\|h\|_{2}^{2}}{\sigma^{2}}+\frac{\|g\|_{3}^{2}\|h\|_{3}}{\sigma^{3}}\right\\}.$

In particular, the bound above can be further simplified as

(2.3) $\sup_{x\in\operatorname{\mathbb{R}}}|P(T_{n}\leq
x)-\Phi(x)|\leq\frac{C(m)}{\sqrt{n}}\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{\sigma^{3}}.$

While the uniform bound in (2.3) resembles the uniform bound for standardized
U-statistics in (1.5), as mentioned in Section 1, it is impossible to obtain a
nonuniform bound of the form in (1.9) that resembles the nonuniform bound in
(1.6) for standardized U-statistics. To see this, we shall first revisit how
Novak (2005, p.342-343) refuted the prospective nonuniform bound for the self-
normalized sum in (1.8), via constructing $X_{1},\dots,X_{n}$ as i.i.d. binary
variables such that

(2.4) $P\Bigl{(}X_{i}=p^{1/2}(1-p)^{-1/2}\Bigr{)}=1-p\text{ and
}P\Bigl{(}X_{i}=-(1-p)^{1/2}p^{-1/2}\Bigr{)}=p$

for some $p\in(0,1)$; the expectation, as well as the second and third
absolute moments of $X_{1}$ is

(2.5)
$\operatorname{\mathbb{E}}[X_{1}]=0,\quad\operatorname{\mathbb{E}}[X_{1}^{2}]=1\text{
and
}\operatorname{\mathbb{E}}[|X_{1}|^{3}]=p^{3/2}(1-p)^{-1/2}+(1-p)^{3/2}p^{-1/2}.$

For such data, by letting

(2.6) $p=p_{n}\equiv n^{-1}\text{ and }x=x_{n}\equiv\sqrt{n}-\epsilon\text{
for any small fixed constant }\epsilon>0,$

the right hand side of (1.8) is seen to be equal to

(2.7)
$C\frac{p_{n}^{\frac{3}{2}}(1-p_{n})^{-\frac{1}{2}}+(1-p_{n})^{\frac{3}{2}}p_{n}^{-\frac{1}{2}}}{\sqrt{n}}d(|x_{n}|)\\\
=C\left\\{n^{-2}(1-n^{-1})^{-1/2}+(1-n^{-1})^{3/2}\right\\}d(|x_{n}|).$

Suppose, towards a contradiction, that the bound in (1.8) _does_ hold.
Consider the event

(2.8)
$\operatorname{\mathcal{E}}_{n}\equiv\\{X_{1}=\dots=X_{n}=p_{n}^{1/2}(1-p_{n})^{-1/2}\\},$

on which the self-normalized sum $S_{n}/V_{n}$ can be easily seen to take upon
the value $\sqrt{n}$, which is greater than $x_{n}$; one can then consequently
derive the lower bound $e^{-1}>0$ for the ”liminf” of the left hand side in
(1.8) as:

$\displaystyle\liminf_{n\rightarrow\infty}\Bigl{[}P(S_{n}/V_{n}\leq
x_{n})-\Phi(x_{n})\Bigr{]}$
$\displaystyle=\liminf_{n\rightarrow\infty}\Bigl{[}P(S_{n}/V_{n}>x_{n})-\bar{\Phi}(x_{n})\Bigr{]}$
$\displaystyle\geq\liminf_{n\rightarrow\infty}\Bigl{[}P(\operatorname{\mathcal{E}}_{n})-\bar{\Phi}(x_{n})\Bigr{]}$
(2.9)
$\displaystyle=\liminf_{n\rightarrow\infty}\Big{[}(1-p_{n})^{n}-\bar{\Phi}(x_{n})\Big{]}=1/e.$

However, this contradicts the presumed bound in (1.8), since the right hand
side in (2.7) converges to zero as $n\rightarrow\infty$, given the assumed
property $\lim_{x\rightarrow\infty}d(x)=0$.

Likewise, the nonuniform Berry-Esseen-type bound (1.9) can’t hold for
Studentized U-statistics either. In fact, assuming the data are as in Novak
(2005)’s construction in (2.4) again, we hereby show that an even wider class
of bounds that include (1.9) as a special case cannot hold: We will show by
contradiction that, it is impossible to have a bound of the form

(2.10) $|P(T_{n}\leq x)-\Phi(x)|\leq C(m,n,x,\mathcal{L}_{X_{1}},h),$

where the right hand side is an absolute term depending only on $m$, $n$, $x$,
the law $\mathcal{L}_{X_{1}}$ of the representative variable $X_{1}$ and
(attributes of) the kernel $h$ in such a way that,

(2.11) $\lim_{x\rightarrow\infty}C(m,n,x,\mathcal{L}_{X_{1}},h)=0\text{ when
the other parameters }(m,n,\mathcal{L}_{X_{1}},h)\text{ are held fixed}.$

First, we define a special real-valued, symmetric kernel $h$ of degree $m\geq
1$ by

(2.12)
$h:\operatorname{\mathbb{R}}^{m}\rightarrow\operatorname{\mathbb{R}}\text{ and
}h(x_{1},\dots,x_{m})\equiv x_{1}+\dots+x_{m}.$

One can check with elementary calculations that
$U_{n}=\frac{m}{n}\sum_{i=1}^{n}X_{i}$ and

$\sum_{i=1}^{n}q_{i}^{2}-nU_{n}^{2}=\bigg{(}\frac{n-m}{n-1}\bigg{)}^{2}\bigg{(}\sum_{i=1}^{n}X_{i}^{2}-n(\bar{X}_{n})^{2}\bigg{)}$

when the U-statistic is formed with this particular kernel in (2.12); as such,
from the definition of $\hat{\sigma}$ in (2.1), one can see that

(2.13) $T_{n}=T_{student}\text{ for any $m\geq 1$ and the kernel in
}\eqref{sum_kernel}.$

Next, recall the classical relationship between the self-normalized sum and
Student’s t-statistics (Efron, 1969):

(2.14)
$T_{student}=\frac{S_{n}}{V_{n}}\Bigg{\\{}\frac{n-1}{n-(S_{n}/V_{n})^{2}}\Bigg{\\}}^{1/2}.$

On the event $\operatorname{\mathcal{E}}_{n}$ in (2.8), $S_{n}/V_{n}$ is equal
to $\sqrt{n}$ and hence $T_{student}$ takes the value $\infty$ in light of
(2.14); as such, by the equality in (2.13),

(2.15) $T_{n}=\infty\text{ on the event }\operatorname{\mathcal{E}}_{n},\text{
for all }n.$

Since
$\lim_{n\rightarrow\infty}P(\operatorname{\mathcal{E}}_{n})=\lim_{n\rightarrow\infty}(1-p_{n})^{n}=e^{-1}$,
we will let $N\in\mathbb{N}$ be such that
$P(\operatorname{\mathcal{E}}_{N})\geq e^{-1}/2$. However, the fact in (2.15)
implies that

$\liminf_{x\rightarrow\infty}(P(T_{N}\leq
x)-\Phi(x))=\liminf_{x\rightarrow\infty}(P(T_{N}>x)-\bar{\Phi}(x))\geq
P(\operatorname{\mathcal{E}}_{N})-\lim_{x\rightarrow\infty}\bar{\Phi}(x)\geq
e^{-1}/2,$

which in turns implies $\liminf_{x\rightarrow\infty}|P(T_{N}\leq
x)-\Phi(x)|\geq e^{-1}/2$. Apparently, the last fact breaks the bound in
(2.10) with the presumed property in (2.11)!

##  3\. Main results

The moral of Novak (2005)’s example in Section 2 is that, due to the way that
the Jackknife Studentizer $\hat{\sigma}$ in (2.1) is constructed, when the
distribution of the data in question is such that $T_{n}$ can take its largest
possible value (i.e. $\infty$) with a non-negligible probability, a bound like
(1.9) may fail to hold. We now state our main theorem, which contains what we
consider to be the correct nonuniform B-E bound for Studentized U-statistics;
it suggests that it is enough to augment the form in (1.9) with an extra term
that decays exponentially in $n$.

######  Theorem 3.1 (Nonuniform B-E bounds for Studentized U-statistics).

Let $X_{1},\dots,X_{n}\in\operatorname{\mathcal{X}}$ be independently and
identically distributed random variables. Under (1.2)-(1.3), $\max(2,m^{2})<n$
and the moment condition $\operatorname{\mathbb{E}}[|h|^{3}]<\infty$, for any
$x\in\operatorname{\mathbb{R}}$, there exist positive absolute constants
$C(m)$, $c_{1}(m)$ and $c_{2}(m)$ such that

(3.1) $|P(T_{n}\leq
x)-\Phi(x)|\leq\exp\Bigg{(}-\frac{c_{1}(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}+\\\
C(m)\Bigg{\\{}\frac{1}{(1+|x|^{3})}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}\sigma^{3}}+\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}\sigma^{3}}\Bigg{)}+\frac{1}{e^{c_{2}(m)|x|}\sqrt{n}}\Bigg{(}\frac{\|g\|_{3}^{2}\|h\|_{3}}{\sigma^{3}}+\frac{\|h\|_{3}^{2}}{\sigma^{2}}\Bigg{)}\Bigg{\\}};$

In particular, this implies, for some positive absolute constants $C(m)$ and
$c(m)$,

(3.2) $|P(T_{n}\leq
x)-\Phi(x)|\leq\exp\Bigg{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}+\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{(1+|x|^{3})\sqrt{n}\sigma^{3}}.$

Theorem 3.1 is proved in Section 4. Note that (3.2) is a simple consequence of
(3.1) because $\|g\|_{3}\leq\|h\|_{3}$, due to the basic U-statistic property
in (4.10) below. For the choice of the probability $p=p_{n}=n^{-1}$ in (2.6),
let us now re-examine Novak (2005)’s binary data in (2.4) and our special
kernel $h$ in (2.12) to demonstrate three features of our new bounds in
Theorem 3.1. Note that, by considering $x_{n}$ in (2.6) and the fact
established in (2.15) for the event $\operatorname{\mathcal{E}}_{n}$ defined
in (2.8), we must have

(3.3) $\liminf_{n\rightarrow\infty}[P(T_{n}\leq
x_{n})-\Phi(x_{n})]=\liminf_{n\rightarrow\infty}[P(T_{n}>x_{n})-\bar{\Phi}(x_{n})]\geq\liminf_{n\rightarrow\infty}[P(\operatorname{\mathcal{E}}_{n})-\Phi(x_{n})]=e^{-1}.$

Moreover, we will also leverage the following moment bounds for the kernel in
(2.12),

(3.4)
$8^{-1}m\operatorname{\mathbb{E}}[|X_{1}|^{3}]\leq\operatorname{\mathbb{E}}[|h|^{3}]\leq
C(m)\operatorname{\mathbb{E}}[|X_{1}|^{3}],$

where $C(m)>0$ is an absolute constant depending only on $m$; the bounds in
(3.4) are direct consequences of the classical Rosenthal’s inequalities
(Rosenthal, 1970, Theorem 3).

  1. (i)

The new B-E bounds can accommodate more ”unusual” data distributions: By the
lower bound
$8^{-1}m\operatorname{\mathbb{E}}[|X_{1}|^{3}]\leq\operatorname{\mathbb{E}}[|h|^{3}]$
in (3.4) and the moment calculations in (2.5), with the choice $p=p_{n}$ in
(2.6), we get

$\displaystyle\exp\Bigg{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}$
$\displaystyle\geq\exp\Bigg{(}-\frac{c(m)n}{(8^{-1}m(n^{-3/2}(1-n^{-1})^{-1/2}+(1-n^{-1})^{3/2}n^{1/2}))^{2}}\Bigg{)}$
$\displaystyle\geq\exp\Bigg{(}-\frac{64\cdot
c(m)}{m^{2}(1-n^{-1})^{3}}\Bigg{)}.$

Hence, given $n>2$, $\exp(-\frac{64\cdot c(m)}{m^{2}(1-n^{-1})^{3}})$ is
larger than the lower bound $e^{-1}$ in (3.3) for a sufficiently small
$c(m)>0$; as such, unlike (2.10), the new bounds (3.1) and (3.2) are not
contradicted.

  2. (ii)

The correction term could be crucial even when $|x|$ is not large relative to
$n$: As
$\operatorname{\mathbb{E}}[|h|^{3}]/\sigma^{3}=n^{-3/2}(1-n^{-1})^{-1/2}+(1-n^{-1})^{3/2}n^{1/2}\sim\sqrt{n}$
as $n\rightarrow\infty$, we have

(3.5)
$\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{(1+|x|^{3})\sqrt{n}\sigma^{3}}\sim\frac{C(m)}{(1+|x|^{3})}\text{
for large }n.$

The last display implies that, for an unusual data distribution where the
moment ratio $\operatorname{\mathbb{E}}[|h|^{3}]/\sigma^{3}$ can be as large
as $\sqrt{n}$, the need for having a correction term as in (3.2) could arise
as long as $x$ is of the order $O(n^{a})$ for even a small $a>0$, because the
term in (3.5) could be already too small to bound the left hand side of (3.2),
as suggested by the lower bound for the ”$\liminf_{n\rightarrow\infty}$” in
(3.3).

  3. (iii)

_The order of $n$ in the correction term is optimal_: Novak (2005)’s example
also illustrates the current order of $n$ in our additive correction proposal
is optimal. Suppose, toward a contradiction, that the correction term had
instead taken the form
$\exp(-\frac{c(m)n^{a}\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}})$
for a power $a>1$, with a faster decay in $n$. In light of the upper bound
$\operatorname{\mathbb{E}}[|h|^{3}]\leq
C(m)\operatorname{\mathbb{E}}[|X_{1}|^{3}]$ in (3.4), such a correction term
could then be further upper bounded by

(3.6)
$\exp\Bigg{(}-\frac{c(m)n^{a-1}}{(n^{-2}(1-n^{-1})^{-1/2}+(1-n^{-1})^{3/2})^{2}}\Bigg{)}$

for the binary data in (2.4), the parameter choice $p=p_{n}=n^{-1}$ in (2.6),
our kernel in (2.12) and a sufficiently small constant $c(m)>0$. Apparently,
because $a-1>0$, the term in the (3.6) converges to zero as
$n\rightarrow\infty$; this implies that the hypothetical correction term with
$a>1$ cannot bound the ”$\liminf_{n\rightarrow\infty}$” in (3.3), because even
its upper bound in (3.6) can’t!

While we believe Theorem 3.1 has nailed down the correct nonuniform B-E bounds
for Studentized U-statistics, two aspects related to the degree $m$ shall
warrant further investigation:

  1. (i)

In the B-E bounds for the _standardized_ U-statistics, Chen and Shao (2007,
Section 3.1) has actually shown that, as $m$ increases, the absolute
constants, i.e. $C_{1}(m)$, $C_{2}(m)$ in (1.5) and (1.6), only grow very
pleasantly at the rate $\sqrt{m}$. However, for Theorem 3.1, due to the
fundamental challenge posed by Studentization, it is unclear to us what the
best possible (i.e. slowest) growth rate of the constants in $m$ should be. We
defer further discussion of this to Section 4.1, after we have finished
proving Theorem 3.1.

  2. (ii)

The condition $\max(2,m^{2})<n$ in Theorem 3.1 is stronger than the typical
$2m<n$ assumed for the uniform B-E bound in Theorem 2.1. As will be seen in
Section 4, letting $\max(2,m^{2})<n$ facilitates our analysis of the lower
tail probability of the Studentizer $\hat{\sigma}$ as a non-negative-kernel
U-statistic using the crucial Lemma 4.3, which ultimately leads to our
correction term with exponential decay in $n$. However, we believe
establishing our theorem under $2m<n$ is potentially feasible, and the related
discussion will appear in Section 4.2.

Aside from our general result in Theorem 3.1, thanks to the delicate _Cram
$\acute{\text{e}}$r-type_ moderate deviation theorem for the self-normalized
sum $S_{n}/V_{n}$ established in Jing et al. (2003), a very refined nonuniform
B-E bound for the Student’s t-statistic, the special case of $T_{n}$ with the
kernel in (1.1), can be established in Theorem 3.2 below; the proof is Section
5. It says that the nonuniform term in $x$ can be further strengthened to be
decaying exponentially in $|x|$. It is an open question whether one can
similarly strengthen the rate of decay in $|x|$ for our general result in
Theorem 3.1, as the current state-of-the-art in _Cram
$\acute{\text{e}}$r-type_ moderate deviation results for Studentized
U-statistics applies to a restricted class of kernels only (Shao and Zhou,
2016, Eqn. $(3.3)$).

######  Theorem 3.2 (Nonuniform B-E bound for Student’s t-statistic).

Let $X_{1},\dots,X_{n}$ be independent and identically distributed real-valued
random variables such that $\operatorname{\mathbb{E}}[X_{1}]=0$,
$0<\operatorname{\mathbb{E}}[X_{1}^{2}]<\infty$ and
$\operatorname{\mathbb{E}}[|X_{1}|^{3}]<\infty$. Assume $n\geq 2$. Then there
exist positive absolute constants $C_{1},C_{2},c_{1},c_{2}>0$ such that

$\bigg{|}P\bigg{(}T_{student}\leq x\bigg{)}-\Phi(x)\bigg{|}\leq
C_{1}\exp\Bigg{(}\frac{-c_{1}n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+\frac{C_{2}}{e^{c_{2}x^{2}}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}.$

The same bound can be stated with $T_{n}$ replaced by the self-normalized sum
$S_{n}/V_{n}$, where $S_{n}$ and $V_{n}$ are defined in (1.7), for possibly
different constants $C_{1},C_{2},c_{1},c_{2}>0$.

##  4\. Proof of the nonuniform B-E bound for Studentized U-statistics

This section lays out the major steps of the proof for Theorem 3.1. It
suffices to consider $x\geq 0$ only, or else one can replace the kernel $h$
with $-h$. Moreover, one can further just focus on $x\geq 1$; for the range
$0\leq x<1$, because $e^{-c(m)x}\geq e^{-c(m)}$ for any small positive
constant $c(m)$, one can always inflate the constant $C(m)$ in (3.1)
sufficiently so that Theorem 3.1 is true by virtue of the uniform bound in
Theorem 2.1. Hence, this section focuses on proving

(4.1) $|P(T_{n}\leq
x)-\Phi(x)|\leq\exp\Bigg{(}-\frac{c_{1}(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}+\\\
C(m)\Bigg{\\{}\frac{1}{(1+x^{3})}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}\sigma^{3}}+\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}\sigma^{3}}\Bigg{)}+\frac{1}{e^{c_{2}(m)x}\sqrt{n}}\Bigg{(}\frac{\|g\|_{3}^{2}\|h\|_{3}}{\sigma^{3}}+\frac{\|h\|_{3}^{2}}{\sigma^{2}}\Bigg{)}\Bigg{\\}}\text{
for }x\geq 1,$

for some absolute constant $C(m),c_{1}(m),c_{2}(m)>0$.

Without loss of generality, we assume

(4.2) $\sigma^{2}=1$

as one can always replace $h(\cdot)$ and $g(\cdot)$ respectively with
$h(\cdot)/\sigma$ and $g(\cdot)/\sigma$ without changing the definition of
$T_{n}$. To prove (4.1), we adopt the framework of _self-normalized_ nonlinear
statistics, which amounts to writing $T_{n}$ as

(4.3) $T_{n}=\frac{W+D_{1}}{(1+D_{2})^{1/2}},$

where

(4.4) $W=W_{n}\equiv\sum_{i=1}^{n}\xi_{i},$

with $\xi_{1},\dots,\xi_{n}$ being independent random variables such that

(4.5) $\operatorname{\mathbb{E}}[{\xi_{i}}]=0\text{ for all
}i=1,\dots,n,\quad{and}\quad\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]=1;$

$D_{1}$ and $D_{2}$ are random ”remainder” terms that are negligible when $n$
is large, with the additional property that

(4.6) $D_{2}\geq-1\text{ almost surely}.$

This is accomplished by first letting

(4.7) $\xi_{i}=\frac{g(X_{i})}{\sqrt{n}}\text{ for }i=1,\dots,n;$

under both assumptions (1.2) and (4.2), it is seen that the properties in
(4.5)are satisfied. Define

(4.8)
$\bar{h}_{k}(x_{1}\dots,x_{k})=h_{k}(x_{1}\dots,x_{k})-\sum_{i=1}^{k}g(x_{i})\text{
for }k=1,\dots,m,$

where

$h_{k}(x_{1},\dots,x_{k})=\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})|X_{1}=x_{1},\dots,X_{k}=x_{k}];$

in particular, $g(x)=h_{1}(x)$,
$h(x_{1},\dots,x_{m})=h_{m}(x_{1},\dots,x_{m})$, and $\bar{h}_{k}$ for any
$k\in\\{1,\dots,m\\}$ has the degeneracy property

(4.9) $\operatorname{\mathbb{E}}[\bar{h}_{k}(X_{1},\dots,X_{k})|X_{i}]=0\text{
for any }i=1,\dots,k.$

For any $p\geq 1$, an important property of the functions $h_{k}$ is that

(4.10)
$\operatorname{\mathbb{E}}\Big{[}|h_{k}|^{p}\Big{]}\leq\operatorname{\mathbb{E}}\Big{[}|h_{k^{\prime}}|^{p}\Big{]}\text{
for }k\leq k^{\prime},$

which necessarily implies, for a constant $C(k)>0$ depending only on $k$,

(4.11) $\operatorname{\mathbb{E}}[|\bar{h}_{k}|^{p}]\leq
C(k)\operatorname{\mathbb{E}}[|h_{k}|^{p}]\leq
C(k)\operatorname{\mathbb{E}}[|h|^{p}].$

By the Hoeffding decomposition, for $W$ in (4.4) constructed with (4.7), we
have

$\frac{\sqrt{n}}{m}U_{n}=W+{n-1\choose m-1}^{-1}\sum_{1\leq
i_{1}<\dots<i_{m}\leq
n}\frac{\bar{h}_{m}(X_{i_{1}},\dots,X_{i_{m}})}{\sqrt{n}}.$

Hence, the ”numerator remainder” for $T_{n}$ can be defined as

(4.12) $D_{1}=D_{1}(X_{1},\dots,X_{n})\equiv{n-1\choose m-1}^{-1}\sum_{1\leq
i_{1}<\dots<i_{m}\leq
n}\frac{\bar{h}_{m}(X_{i_{1}},\dots,X_{i_{m}})}{\sqrt{n}},$

and the ”denominator remainder” can be taken as

(4.13) $D_{2}=D_{2}(X_{1},\dots,X_{m})\equiv\hat{\sigma}^{2}-1.$

By further defining

$\quad
V_{n}^{2}=\sum_{i=1}^{n}\xi_{i}^{2},\quad\Psi_{n,i}=\sum_{\begin{subarray}{c}1\leq
i_{1}<\dots<i_{m-1}\leq n\\\ i_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}\frac{\bar{h}_{m}(X_{i},X_{i_{1}},\dots,X_{i_{m-1}})}{\sqrt{n}}\quad\text{
and }\quad\Lambda_{n}^{2}=\sum_{i=1}^{n}\Psi_{n,i}^{2},$

as well as

(4.14)
$\delta_{1}^{*}=\delta_{1n}^{*}\equiv\left[\frac{n(m-1)^{2}}{(n-m)^{2}}+\frac{2(m-1)}{(n-m)}\right]W^{2}+\frac{(n-1)^{2}}{{n-1\choose
m-1}^{2}(n-m)^{2}}\Lambda_{n}^{2}+\frac{2(n-1)(m-1)}{(n-m)^{2}{n-1\choose
m-1}}\sum_{i=1}^{n}W\Psi_{n,i},$

one can then also write $D_{2}$ as

(4.15) $D_{2}=d_{n}^{2}(V_{n}^{2}+\delta_{1}+\delta_{2})-1\text{ for
}d_{n}=\sqrt{\frac{n}{n-1}},$

with

(4.16) $\delta_{1}\equiv\delta_{1}^{*}-\frac{(n-1)^{2}}{(n-m)^{2}}U_{n}^{2}$

and

$\delta_{2}\equiv\frac{2(n-1)}{(n-m)}{n-1\choose
m-1}^{-1}\sum_{i=1}^{n}\xi_{i}\Psi_{n,i};$

see our related work Leung and Shao (2023, Section 3) for the derivation of
(4.15).333Specifically, it was showed that
${\hat{\sigma}^{*}}^{2}=d_{n}^{2}(V_{n}^{2}+\delta_{1}^{*}+\delta_{2})$; see
the self-normalized U-statistic in (4.43). One can then deduce from (2.1) that
$\hat{\sigma}^{2}=d_{n}^{2}(V_{n}^{2}+\delta_{1}^{*}-\frac{(n-1)^{2}}{(n-m)^{2}}U_{n}^{2}+\delta_{2})$.
We also note that, although defining $V_{n}^{2}=\sum_{i=1}^{n}\xi_{i}^{2}$
slightly abuses the definition of $V_{n}^{2}$ for the self-normalized sum in
(1.7), one can think of $\xi_{i}$’s as analogous to the real-valued $X_{i}$’s
in the self-normalized sum.

In Leung and Shao (2023), replacing the more common truncation technique,
_variable censoring_ is advocated as the appropriate device to establish B-E
bounds for self-normalized nonlinear statistics under the Stein-method
approach. In this paper, variable censoring is also adopted; in particular,
the censored summands

$\xi_{b,i}\equiv\xi_{i}I(|\xi_{i}|\leq 1)+I(\xi_{i}>1)-I(\xi_{i}<-1)\text{ for
each }i=1,\dots,n,$

as well as their sum

$W_{b}\equiv\sum_{i=1}^{n}\xi_{b,i}$

will also figure in our proof. However, the other two remainder terms $D_{1}$
and $D_{2}$ have to be censored in a considerably more delicate manner as
described next. First, we shall define a special positive constant
”$\mathfrak{c}_{m}$” via its square:

(4.17)
$\mathfrak{c}_{m}^{2}\equiv\Big{(}1-\frac{m^{2}}{m^{2}+1}\Big{)}\times\mathfrak{b}_{m},$

where $\mathfrak{b}_{m}$ is a constant depending also only on $m$ defined as

(4.18) $\mathfrak{b}_{m}\equiv\begin{cases}&\frac{1}{2}\text{ if }m=1\text{ or
}2;\\\
&\underbrace{\left(\frac{m}{2m-2}\right)\cdot\left(\frac{m-1}{2m-3}\right)\cdots\left(\frac{4}{m+2}\right)\cdot\left(\frac{3}{m+1}\right)}_{(m-2)\text{
many terms }}\text{ if }m\geq 3.\end{cases}$

Later, it will become clear later why $\mathfrak{c}_{m}$ is defined in this
specific way; for now, it is enough to know that $\mathfrak{c}_{m}$ only
depends on $m$ and has the property that

$0<\mathfrak{c}_{m}<1.$

The censored version of the numerator remainder $D_{1}$ is defined to be

(4.19)
$\bar{D}_{1,x}=D_{1}I\left(|D_{1}|\leq\frac{\mathfrak{c}_{m}x}{4}\right)+\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1}>\frac{\mathfrak{c}_{m}x}{4}\right)-\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1}<-\frac{\mathfrak{c}_{m}x}{4}\right).$

For the denominator remainder, replacing certain $\xi_{i}$’s with
$\xi_{b,i}$’s in (4.15) we first define

$D_{2,V_{b},\delta_{1},\delta_{2,b}}=d_{n}^{2}(V_{b}^{2}+\delta_{1}+\delta_{2,b})-1.$

where

(4.20) $V^{2}_{b}=V^{2}_{n,b}\equiv\sum_{i=1}^{n}\xi_{b,i}^{2}\text{ and
}\delta_{2,b}\equiv\frac{2(n-1)}{(n-m)}{n-1\choose
m-1}^{-1}\sum_{i=1}^{n}\xi_{b,i}\Psi_{n,i}$

We further censor $\delta_{1}$ and $\delta_{2,b}$ as

(4.21) $\bar{\delta}_{1}=\delta_{1}I(|\delta_{1}|\leq
n^{-1/2})+n^{-1/2}I(\delta_{1}>n^{-1/2})-n^{-1/2}I(\delta_{1}<-n^{-1/2})$

and

$\bar{\delta}_{2,b}=\delta_{2,b}I(|\delta_{2,b}|\leq
1)+I(\delta_{2,b}>1)-I(\delta_{2,b}<-1),$

and define

(4.22)
$D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}=d_{n}^{2}(V_{b}^{2}+\bar{\delta}_{1}+\bar{\delta}_{2,b})-1.$

Finally, we censor $D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}$ as

(4.23) $\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\equiv
D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq
D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\leq 1\Bigg{)}\\\
+I\Bigg{(}D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}.$

With these censoring constructions, now we start to prove (4.1): First rewrite

$P(T_{n}>x)=P(W+D_{1}>x(1+D_{2})^{1/2})$

and define the events:

$\displaystyle\operatorname{\mathcal{E}}_{1}$
$\displaystyle\equiv\left\\{W+D_{1}>x(1+D_{2})^{1/2},\;\
|D_{1}|>\frac{\mathfrak{c}_{m}x}{4}\right\\}\bigcup\left\\{W+D_{1}>x(1+D_{2})^{1/2},\;\
D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\right\\};$
$\displaystyle\operatorname{\mathcal{E}}_{2}$
$\displaystyle\equiv\left\\{W+\bar{D}_{1,x}>x\Bigg{(}1+\max\bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2}\bigg{)}\Bigg{)}^{1/2},\;\max_{1\leq
i\leq n}|\xi_{i}|>1\right\\};$ $\displaystyle\operatorname{\mathcal{E}}_{3}$
$\displaystyle\equiv\left\\{W_{b}+\bar{D}_{1,x}>x\Bigg{(}1+\max\bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2,V_{b},\delta_{1},\delta_{2,b}}\bigg{)}\Bigg{)}^{1/2},\;\
|\delta_{1}|>\frac{1}{\sqrt{n}}\right\\}$ $\displaystyle\hskip
56.9055pt\bigcup\left\\{W_{b}+\bar{D}_{1,x}>x\Bigg{(}1+\max\bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2,V_{b},\delta_{1},\delta_{2,b}}\bigg{)}\Bigg{)}^{1/2},\;\
|\delta_{2,b}|>1\right\\};$ $\displaystyle\operatorname{\mathcal{E}}_{4}$
$\displaystyle\equiv\left\\{W_{b}+\bar{D}_{1,x}>x\Bigg{(}1+\max\bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\bigg{)}\Bigg{)}^{1/2},\;\
|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right\\}.$

The following sequence of inclusions are then seen to hold by progressively
using

$\tilde{\operatorname{\mathcal{E}}}_{\ell}\equiv\cup_{i=1}^{\ell}\operatorname{\mathcal{E}}_{i},\quad\ell=1,\dots,4,$

as covering events:

$\\{T_{n}>x\\}\backslash\operatorname{\mathcal{E}}_{1}\subset\Bigg{\\{}P\Big{(}W+\bar{D}_{1,x}>x\Big{(}1+\max\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2}\Big{)}\Big{)}^{1/2}\Big{)}\Bigg{\\}}\subset\\{T_{n}>x\\}\cup\operatorname{\mathcal{E}}_{1}$
$\downarrow$
$\\{T_{n}>x\\}\backslash\tilde{\operatorname{\mathcal{E}}}_{2}\subset\Bigg{\\{}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\max\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2,V_{b},\delta_{1},\delta_{2,b}}\Big{)}\Big{)}^{1/2}\Big{)}\Bigg{\\}}\subset\\{T_{n}>x\\}\cup\tilde{\operatorname{\mathcal{E}}}_{2}$
$\downarrow$
$\\{T_{n}>x\\}\backslash\tilde{\operatorname{\mathcal{E}}}_{3}\subset\Bigg{\\{}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\max\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1,D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\Big{)}\Big{)}^{1/2}\Big{)}\Bigg{\\}}\subset\\{T_{n}>x\\}\cup\tilde{\operatorname{\mathcal{E}}}_{3}$
$\downarrow$ (4.24)
$\\{T_{n}>x\\}\backslash\tilde{\operatorname{\mathcal{E}}}_{4}\subset\Bigg{\\{}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\Big{)}^{1/2}\Big{)}\Bigg{\\}}\subset\\{T_{n}>x\\}\cup\tilde{\operatorname{\mathcal{E}}}_{4}.$

The last event inclusion (4.24) implies the inequality

(4.25) $\Big{|}P(T_{n}>x)-\bar{\Phi}(x)\Big{|}\leq\\\
R_{x}+P\left(D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\right)+\Big{|}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\Big{)}^{1/2}\Big{)}-\bar{\Phi}(x)\Big{|},$

where

$R_{x}\equiv P\Bigg{(}W+D_{1}>x(1+D_{2})^{1/2},\;\
|D_{1}|>\frac{\mathfrak{c}_{m}x}{4}\Bigg{)}+\sum_{i=2}^{4}P(\operatorname{\mathcal{E}}_{i})$

is the sum of the probabilities of all covering events except

$\Big{\\{}W+D_{1}>x(1+D_{2})^{1/2},\;\
D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{\\}}$

whose probability can be bounded by
$P(D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1)$. Hence, the proof boils down to
proving bounds for the three terms on the right hand side of (4.25). We first
state the bound for the ”$x$-dependent” term $R_{x}$. In light of the
inequality:

$x\Bigg{(}1+\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}\Bigg{)}^{1/2}-\bar{D}_{1,x}\geq
x\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}\Big{)}^{1/2}-\frac{\mathfrak{c}_{m}x}{4}=\frac{\mathfrak{c}_{m}x}{2}$

which is true by the definition of $\bar{D}_{1,x}$, it can be seen that

(4.26) $R_{x}\leq
P\left(|D_{1}|>\frac{\mathfrak{c}_{m}x}{4}\right)+P\left(W\geq\frac{\mathfrak{c}_{m}x}{2},\max_{1\leq
i\leq
n}|\xi_{i}|>1\right)+P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{1}|>\frac{1}{\sqrt{n}}\right)\\\
+P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{2,b}|>1\right)+P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right),$

leading us to the following bound:

######  Lemma 4.1 (Nonuniform bound for $R_{x}$).

For $x\geq 1$, assuming (1.2) and (4.2), we have the following bounds of rate
no larger than $1/\sqrt{n}$:

  1. (i)

$P\left(|D_{1}|>\frac{\mathfrak{c}_{m}x}{4}\right)\leq\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{\mathfrak{c}_{m}^{3}n^{3/2}(1+x^{3})}$;

  2. (ii)

$P\left(W\geq\frac{\mathfrak{c}_{m}x}{2},\max_{1\leq i\leq
n}|\xi_{i}|>1\right)\leq\frac{C\operatorname{\mathbb{E}}[|g|^{3}]}{\mathfrak{c}_{m}^{3}(1+x^{3})\sqrt{n}}$;

  3. (iii)

$P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{1}|>\frac{1}{\sqrt{n}}\right)\leq
C(m)e^{-\mathfrak{c}_{m}x/2}\frac{\|h\|_{3}^{2}}{\sqrt{n}}$;

  4. (iv)

$P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{2,b}|>1\right)\leq
C(m)e^{-\mathfrak{c}_{m}x/2}\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}$;

  5. (v)

$P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right)\leq
Ce^{-\mathfrak{c}_{m}x/2}\Big{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]+\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Big{)}$.

In particular, via (4.26) these bounds together imply

(4.27)
$R_{x}\leq\frac{C_{1}(m)}{(1+x^{3})}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}+\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}\Bigg{)}+\frac{C_{2}(m)}{e^{\mathfrak{c}_{m}x/2}}\Bigg{(}\frac{\|h\|_{3}^{2}}{\sqrt{n}}\Bigg{)},$

for some absolute constants $C_{1}(m),C_{2}(m)>0$.

The proof of Lemma 4.1 in Appendix B follows fairly standard arguments, and we
note that the proofs for $(iii)$-$(v)$ repeatedly use the Chernoff-type bound
$I\Big{(}W_{b}\geq\frac{\mathfrak{c}_{m}x}{2}\Big{)}\leq
e^{W_{b}-\frac{\mathfrak{c}_{m}x}{2}}$ to result in the exponentially
nonuniform terms in $x$. Next, we bound the other $x$-dependent term
$|P(W_{b}+\bar{D}_{1,x}>x(1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}})^{1/2})-\bar{\Phi}(x)|$
from (4.25) in Lemma 4.2 below. Its proof in Appendix C involves Stein’s
method, and is largely similar to the proof of the uniform B-E bounds in Leung
and Shao (2023), except that the properties of the solution to the Stein
equation is more thoroughly exploited, to ensure the nonuniformity in $x$ of
the bound.

######  Lemma 4.2 (Intermediate nonuniform bound by Stein’s method).

For $x\geq 1$, there exist absolute constants $C(m),c(m)>0$ depending only on
$m$ such that

(4.28)
$\bigg{|}P\bigg{(}W_{b}+\bar{D}_{1,x}>x\big{(}1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}\big{)}^{1/2}\bigg{)}-\bar{\Phi}(x)\bigg{|}\leq\frac{C(m)}{e^{c(m)x}}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]+\|g\|_{3}^{2}\|h\|_{3}}{\sqrt{n}}\Bigg{)}.$

To summarize Lemmas 4.1 and 4.2 in a nutshell: The delicate internal/external
censoring operations applied to the terms $W$, $D_{1}$ and $D_{2}$ allow for a
desired nonuniform bound of rate $1/\sqrt{n}$ to be established for
$|P(W_{b}+\bar{D}_{1,x}>x(1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}})^{1/2})-\bar{\Phi}(x)|$
under minimal moment conditions, while ensuring that a bound depending on $x$
can be established for $R_{x}$ by way of the inequality in (4.26). We also
remark that the crude censoring techniques in Leung and Shao (2023) are
insufficient to prove a nonuniform bound since they would have severed the
dependence on $x$.

With (4.25), (4.27) and (4.28), to finish proving (4.1) under (4.2), it
remains to show, for a small constant $c(m)>0$ depending only on $m$, the
exponential lower bound

(4.29)
$P\Big{(}D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}=P\Big{(}1+D_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}\Big{)}\leq\exp\Bigg{(}-\frac{c(m)n}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}$

for $D_{2}$ in (4.15). The key observation is that, overall,
$\hat{\sigma}^{2}$ can be understood as a U-statistic constructed with a _non-
negative_ kernel. First, write

$\displaystyle\sum_{i=1}^{n}q_{i}^{2}$
$\displaystyle=\sum_{i=1}^{n}\left({n-1\choose
m-1}^{-1}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{m-1}\leq n\\\
i_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}h(X_{i},X_{i_{1}},\dots,X_{i_{m-1}})\right)^{2}$
$\displaystyle={n-1\choose
m-1}^{-2}\sum_{i=1}^{n}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{m-1}\leq
n\\\ 1\leq j_{1}<\cdots<j_{m-1}\leq n\\\ i_{l},j_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}h(X_{i},X_{i_{1}},\dots,X_{i_{m-1}})h(X_{i},X_{j_{1}},\dots,X_{j_{m-1}})$
$\displaystyle={n-1\choose m-1}^{-2}\left(\sum_{k=m}^{2m-1}\sum_{1\leq
i_{1}<\dots<i_{k}\leq
n}\tilde{\operatorname{\mathcal{H}}}_{k}(X_{i_{1}},\dots,X_{i_{k}})\right)$
$\displaystyle={n-1\choose m-1}^{-2}\sum_{1\leq i_{1}<\dots<i_{2m}\leq
n}\tilde{\mathfrak{h}}(X_{i_{1}},\dots,X_{i_{2m}}),$

where
$\tilde{\operatorname{\mathcal{H}}}_{k}:\mathbb{R}^{k}\longrightarrow\mathbb{R}$
is a symmetric kernel of degree $k$ induced by $h(\cdot)$ defined as

(4.30)
$\tilde{\operatorname{\mathcal{H}}}_{k}(x_{1},\dots,x_{k})\equiv(2m-k)\times\sum_{\begin{subarray}{c}\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,\dots,k\\}:\\\
|\operatorname{\mathcal{S}}_{1}|=2m-k\\\
|\operatorname{\mathcal{S}}_{2}|=|\operatorname{\mathcal{S}}_{3}|=k-m\\\
\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\text{disjoint}\end{subarray}}h(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{2}})h(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{3}}),\\\
\text{ for each }k=m,\dots,2m-1,$

and $\tilde{\mathfrak{h}}$ is the symmetric kernel of degree $2m$ further
derived from (4.30) defined as

(4.31)
$\tilde{\mathfrak{h}}(x_{1},\dots,x_{2m})\equiv\sum_{k=m}^{2m-1}{n-k\choose
2m-k}^{-1}\sum_{1\leq l_{1}<\dots<l_{k}\leq
2m}\tilde{\operatorname{\mathcal{H}}}_{k}(x_{l_{1}},\dots,x_{l_{k}}).$

Next, upon expansion,

$\displaystyle U_{n}^{2}$ $\displaystyle={n\choose
m}^{-2}\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{m}\leq n\\\ 1\leq
j_{1}<\cdots<j_{m}\leq
n\end{subarray}}h(X_{i_{1}},\dots,X_{i_{m}})h(X_{j_{1}},\dots,X_{j_{m}})$
$\displaystyle={n\choose m}^{-2}\left(\sum_{k=m}^{2m}\sum_{1\leq
i_{1}<\dots<i_{k}\leq
n}\breve{\operatorname{\mathcal{H}}}_{k}(X_{i_{1}},\dots,X_{i_{k}})\right)$
$\displaystyle={n\choose m}^{-2}\sum_{1\leq i_{1}<\dots<i_{2m}\leq
n}\breve{\mathfrak{h}}(X_{i_{1}},\dots,X_{i_{2m}}),$

where
$\breve{\operatorname{\mathcal{H}}}_{k}:\mathbb{R}^{k}\longrightarrow\mathbb{R}$
is a symmetric kernel of degree $k$ induced by $h(\cdot)$ defined as

(4.32)
$\breve{\operatorname{\mathcal{H}}}_{k}(x_{1},\dots,x_{k})\equiv\sum_{\begin{subarray}{c}\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,\dots,k\\}:\\\
|\operatorname{\mathcal{S}}_{1}|=2m-k\\\
|\operatorname{\mathcal{S}}_{2}|=|\operatorname{\mathcal{S}}_{3}|=k-m\\\
\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\text{disjoint}\end{subarray}}h(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{2}})h(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{3}})\text{,
for each }k=m,\dots,2m,$

and $\breve{\mathfrak{h}}$ is the symmetric kernel of degree $2m$ further
derived from (4.32) defined as

(4.33)
$\breve{\mathfrak{h}}(x_{1},\dots,x_{2m})\equiv\sum_{k=m}^{2m}{n-k\choose
2m-k}^{-1}\sum_{1\leq l_{1}<\dots<l_{k}\leq
2m}\breve{\operatorname{\mathcal{H}}}_{k}(x_{l_{1}},\dots,x_{l_{k}}).$

With the above expressions for $\sum_{i=1}^{n}q_{i}^{2}$ and $U_{n}^{2}$ both
as U-statistics of degree $2m$, from (2.1), one can write

(4.34) $\hat{\sigma}^{2}=A(m,n)\frac{\sum_{1\leq i_{1}<\dots<i_{2m}\leq
n}\mathfrak{h}(X_{i_{1}},\dots,X_{i_{2m}})}{{n\choose 2m}},$

where

(4.35) $A(n,m)\equiv\frac{n-1}{(n-m)^{2}(n-2m+1)}{n\choose 2m}{n-1\choose
m-1}^{-2}$

and

(4.36)
$\mathfrak{h}(x_{1},\dots,x_{2m})\equiv(n-2m+1)\Bigg{\\{}\tilde{\mathfrak{h}}(x_{1},\dots,x_{2m})-\frac{m^{2}}{n}\breve{\mathfrak{h}}(x_{1},\dots,x_{2m})\Bigg{\\}};$

Hence, up to the multiplicative factor $A(n,m)$, $\hat{\sigma}^{2}$ is a
U-statistic of degree $2m$. Moreover, it is not hard to see that

(4.37) $\mathfrak{h}(x_{1},\dots,x_{2m})\geq 0\text{ for all values of
$x_{1},\dots,x_{2m}$};$

when $n=2m$, from the original definition of $\hat{\sigma}^{2}$ in (2.1), it
is seen that, irrespective of the values of $X_{1},\dots,X_{2m}$,

$A(m,n)\mathfrak{h}(X_{1},\dots,X_{2m})={\hat{\sigma}}^{2}=\frac{n-1}{(n-m)^{2}}\sum_{i=1}^{n}(q_{i}-U_{n})^{2}\geq
0,$

so $\mathfrak{h}$ can only take on non-negative value since $A(m,n)>0$.

With the insights above, we are primed to leverage the following exponential
lower tail bound for non-negative kernel U-statistics to develop the
exponential bound in (4.29). This result is of independent interest, and it
naturally extends a known exponential lower tail bound for a sum of
independent non-negative variables in the literature (de la Peña et al., 2009,
Theorem 2.19); surprisingly, we could not locate a result similar to Lemma 4.3
elsewhere. Its proof is included in Appendix D, which uses a well-known trick
by Hoeffding (1963).

######  Lemma 4.3 (Exponential lower tail bound for U-statistics with non-
negative kernels).

Assume that $U_{n}={n\choose m}^{-1}\sum_{1\leq i_{1}<\dots<i_{m}\leq
n}h(X_{i_{1}},\dots,X_{i_{m}})$ is a U-statistic of degree $m$, and
$h:\operatorname{\mathcal{X}}^{m}\longrightarrow\operatorname{\mathbb{R}}_{\geq
0}$ can only take non-negative values, with the property that
$\operatorname{\mathbb{E}}[h^{p}(X_{1},\dots,X_{m})]<\infty$ for some
$p\in(1,2]$. Then for $0<x\leq\operatorname{\mathbb{E}}[h]$,

$P(U_{n}\leq
x)\leq\exp\left(\frac{-[n/m](p-1)(\operatorname{\mathbb{E}}[h]-x)^{p/(p-1)}}{p(\operatorname{\mathbb{E}}[h^{p}])^{1/(p-1)}}\right),$

where $[n/m]$ is defined as the greatest integer less than $n/m$.

Since

$\frac{n((n-m-1)!)^{2}}{(n-2)!(n-2m+1)!}=\begin{cases}&\frac{n}{n-m}\text{ if
}m=1\text{ or }2;\\\
&\frac{n}{n-2}\times\underbrace{\big{(}\frac{n-m-1}{n-3}\big{)}\cdot\big{(}\frac{n-m-2}{n-4}\big{)}\cdots\big{(}\frac{n-2m+2}{n-m}\big{)}}_{m-2\text{
many terms}}\text{ if }m\geq 3,\end{cases}$

in light of the assumption that $n>\max(2,m^{2})$ (which implies $n>2m$) and
the definition of $\mathfrak{b}_{m}$ in (4.18), one can derive the lower bound

(4.38)
$A(n,m)=\frac{\\{(m-1)!\\}^{2}}{(2m)!}\frac{n((n-m-1)!)^{2}}{(n-2)!(n-2m+1)!}\geq\frac{\\{(m-1)!\\}^{2}}{(2m)!}\mathfrak{b}_{m}.$

Moreover, because for any disjoint subsets
$\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,\dots,k\\}$
such that $|\operatorname{\mathcal{S}}_{1}|=2m-k$,
$|\operatorname{\mathcal{S}}_{2}|=|\operatorname{\mathcal{S}}_{3}|=k-m$ ,

$\operatorname{\mathbb{E}}[h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})]=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})|X_{\operatorname{\mathcal{S}}_{1}}]^{2}]=\operatorname{\mathbb{E}}[h_{2m-k}^{2}(X_{1},\dots,X_{2m-k})],$

we have

$\operatorname{\mathbb{E}}[\tilde{\operatorname{\mathcal{H}}}_{k}]=(2m-k){k\choose
2m-k}{2k-2m\choose k-m}\operatorname{\mathbb{E}}[h_{2m-k}^{2}]\text{ and
}\operatorname{\mathbb{E}}[\breve{\operatorname{\mathcal{H}}_{k}}]={k\choose
2m-k}{2k-2m\choose k-m}\operatorname{\mathbb{E}}[h_{2m-k}^{2}].$

As such, the expectation of $\mathfrak{h}$ can be computed as

$\displaystyle\operatorname{\mathbb{E}}[\mathfrak{h}]$
$\displaystyle=(n-2m+1)\Bigg{\\{}\operatorname{\mathbb{E}}[\tilde{\mathfrak{h}}]-\frac{m^{2}}{n}\operatorname{\mathbb{E}}[\breve{\mathfrak{h}}]\Bigg{\\}}$
$\displaystyle=(n-2m+1)\Bigg{\\{}\sum_{k=m}^{2m-1}{n-k\choose
2m-k}^{-1}{2m\choose
k}\operatorname{\mathbb{E}}[\tilde{\operatorname{\mathcal{H}}}_{k}]-\frac{m^{2}}{n}\sum_{k=m}^{2m}{n-k\choose
2m-k}^{-1}{2m\choose
k}\operatorname{\mathbb{E}}[\breve{\operatorname{\mathcal{H}}_{k}}]\Bigg{\\}}$
$\displaystyle=(n-2m+1)\sum_{k=m}^{2m-1}{n-k\choose 2m-k}^{-1}{2m\choose
k}{k\choose 2m-k}{2k-2m\choose
k-m}\Big{(}2m-k-\frac{m^{2}}{n}\Big{)}\operatorname{\mathbb{E}}[h_{2m-k}^{2}]$
$\displaystyle=(n-2m+1)\sum_{k=m}^{2m-1}\frac{(2m)!}{\\{(2m-k)!(k-m)!\\}^{2}}{n-k\choose
2m-k}^{-1}\Big{(}2m-k-\frac{m^{2}}{n}\Big{)}\operatorname{\mathbb{E}}[h_{2m-k}^{2}]$
(4.39)
$\displaystyle=\Big{(}1-\frac{m^{2}}{n}\Big{)}\frac{(2m)!}{\\{(m-1)!\\}^{2}}+\sum_{k=m}^{2m-2}\frac{(2m)!(n-2m+1)}{\\{(2m-k)!(k-m)!\\}^{2}}{n-k\choose
2m-k}^{-1}\Big{(}2m-k-\frac{m^{2}}{n}\Big{)}\operatorname{\mathbb{E}}[h_{2m-k}^{2}],$

where the third equality uses that
$\operatorname{\mathbb{E}}[h_{0}^{2}]=\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})h(X_{m+1},\dots,X_{2m})]=0$,
and the last equality uses
$\operatorname{\mathbb{E}}[h_{1}^{2}]=\operatorname{\mathbb{E}}[g^{2}]=1$.
Under our assumption $n>m^{2}$, because the quantities $(1-m^{2}/n)$ and
$(2m-k-m^{2}/n)$ for all $k=m,\dots,2m-2$ are positive, all the summands in
(4.39) are positive. In particular, this implies

(4.40)
$\operatorname{\mathbb{E}}[\mathfrak{h}]\geq\frac{(2m)!}{\\{(m-1)!\\}^{2}}\Big{(}1-\frac{m^{2}}{n}\Big{)}\geq\frac{(2m)!}{\\{(m-1)!\\}^{2}}\Big{(}1-\frac{m^{2}}{m^{2}+1}\Big{)}.$

Hence, with the lower bound for $A(m,n)$ in (4.38),

$\displaystyle
P\Big{(}{\hat{\sigma}}^{2}\leq\frac{9\mathfrak{c}_{m}^{2}}{16}\Big{)}$
$\displaystyle\leq P\Bigg{(}\frac{\sum_{1\leq i_{1}<\dots<i_{2m}\leq
n}\mathfrak{h}(X_{i_{1}},\dots,X_{i_{2m}})}{{n\choose
2m}}\leq\frac{9}{16}\cdot\frac{(2m)!}{\\{(m-1)!\\}^{2}}\Big{(}1-\frac{m^{2}}{m^{2}+1}\Big{)}\Bigg{)}$
(4.41)
$\displaystyle\leq\exp\left(-\frac{[n/m]\Big{\\{}\frac{7(2m)!}{16\\{(m-1)!\\}^{2}}\Big{(}1-\frac{m^{2}}{m^{2}+1}\Big{)}\Big{\\}}^{3}}{3(\operatorname{\mathbb{E}}[\mathfrak{h}^{3/2}])^{2}}\right)$

where the last inequality comes from applying Lemma 4.3 to ${n\choose
2m}^{-1}\sum\mathfrak{h}$ by taking
$x=\frac{9(2m)!}{16\\{(m-1)!\\}^{2}}(1-\frac{m^{2}}{m^{2}+1})$ and $p=3/2$,
using the kernel non-negativity in (4.37) and the kernel mean lower bound in
(4.40). The following moment bound for centered U-statistics proved in
Appendix D can be used to further understand
$\operatorname{\mathbb{E}}[|\mathfrak{h}|^{3/2}]$.

######  Lemma 4.4 (General moment bound of U-statistics).

Suppose $h(x_{1},\dots,x_{m})$ is a real-valued symmetric kernel, with
$\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})]=0$ and
$\operatorname{\mathbb{E}}[|h(X_{1},\dots,X_{m})|^{p}]<\infty$ for some
$p\in[1,\infty)$. Let $r\geq 1$ be the order of degeneracy of $U_{n}$, i.e.
$r$ is the first integer for which, as functions,

$h_{k}(x_{1},\dots,x_{k})=0\text{ for }k=1,\dots,r-1,\text{ and }\quad
h_{r}(x_{1},\dots,x_{r})\neq 0.$

For positive constants $C(m,r,p)>0$, we have

(4.42)
$\operatorname{\mathbb{E}}[|U_{n}|^{p}]\leq\begin{cases}\frac{C(m,r,p)\operatorname{\mathbb{E}}[|h|^{p}]}{n^{r(p-1)}}&\text{if
}1\leq p\leq 2;\\\
\frac{C(m,r,p)\operatorname{\mathbb{E}}[|h|^{p}]}{n^{(rp)/2}}&\text{if }p\geq
2.\end{cases}$

With Lemma 4.4, by the Cauchy inequality

$\operatorname{\mathbb{E}}[|h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|^{3/2}]\leq\sqrt{\operatorname{\mathbb{E}}[|h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|^{3}]}\sqrt{\operatorname{\mathbb{E}}[|h(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|^{3}]}=\operatorname{\mathbb{E}}[|h|^{3}],$

for any
$\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,\dots,k\\}$,
one can derive

$\displaystyle\operatorname{\mathbb{E}}[|\mathfrak{h}|^{3/2}]$
$\displaystyle\leq
Cn^{3/2}(\operatorname{\mathbb{E}}[|\tilde{\mathfrak{h}}|^{3/2}]+\frac{m^{3}}{n^{3/2}}\operatorname{\mathbb{E}}[|\breve{\mathfrak{h}}|^{3/2}])$
$\displaystyle\leq C(m)n^{3/2}\Bigg{(}\sum_{k=m}^{2m-1}{n-k\choose
2m-k}^{-3/2}\operatorname{\mathbb{E}}[|\tilde{\operatorname{\mathcal{H}}}_{k}|^{3/2}]+n^{-3/2}\sum_{k=m}^{2m}{n-k\choose
2m-k}^{-3/2}\operatorname{\mathbb{E}}[|\breve{\operatorname{\mathcal{H}}}_{k}|^{3/2}]\Bigg{)}$
$\displaystyle\leq C(m)n^{3/2}\Bigg{(}\sum_{k=m}^{2m-1}{n-k\choose
2m-k}^{-3/2}\operatorname{\mathbb{E}}[|h|^{3}]+n^{-3/2}\sum_{k=m}^{2m}{n-k\choose
2m-k}^{-3/2}\operatorname{\mathbb{E}}[|h|^{3}]\Bigg{)}$
$\displaystyle=C(m)\operatorname{\mathbb{E}}[|h|^{3}],$

which can then conclude (4.29) from (4.41).

###  4.1. How the constants scale in $m$

The special constants $\mathfrak{c}_{m}$ and $\mathfrak{b}_{m}$ defined in
(4.17) and (4.18) play critical roles in arriving at our lower tail bound for
$\hat{\sigma}^{2}$ in (4.41), which ultimately induces the correction term
with exponential decay in $n$, in the final B-E bounds (3.1) and (3.2). For
$m\geq 3$, by Stirling’s formula, we get that

$\mathfrak{b}_{m}=\frac{(m!)^{2}}{2\cdot(2m-2)!}\sim\frac{2\sqrt{\pi}}{e^{2}4^{m}}\cdot\Big{(}\frac{m}{m-1}\Big{)}^{2m}\cdot
m\cdot(m-1)^{3/2},$

where ”$\sim$” means their ratio tends to $1$ as $m\rightarrow\infty$. As
such, $\mathfrak{b}_{m}$, and therefore $\mathfrak{c}_{m}$, decays exponential
in $m$, due to the factor $4^{m}$ on the right hand side of the prior display.
By observing where the constant $\mathfrak{c}_{m}$ figures in the inequalities
of Lemma 4.1$(i)-(v)$, our proof methodology indicates that the big constants
appearing in (3.1) and (3.2), denoted by $C(m)$, could potentially grow
exponentially in $m$! This is in stark contrast to the B-E bounds of the
_standardized_ U-statistics in (1.5) and (1.6), where the constants are known
to only scale like $\sqrt{m}$ (Chen and Shao, 2007, Theorem 3.1). It is not
clear to us whether there could be a different proof that can bring down the
order of these constants in $m$. Note that, $\mathfrak{b}_{m}$ and
$\mathfrak{c}_{m}$ are direct by-products of the factor $A(n,m)$ from
analyzing the Studentizer $\hat{\sigma}$ with the tight exponential lower tail
bound of Lemma 4.3, and $A(n,m)$ is in itself intrinsic to the structure of
the Studentizer $\hat{\sigma}$ as seen in (4.34). As such, the possible
exponential dependence on $m$ of our constants in Theorem 3.1 could well be a
unique, perhaps undesirable, nature of Studentized U-statistics.

###  4.2. The required sample size relative to $m$

In our proof above, we have effectively used the assumed condition $m^{2}<n$
to demonstrate that all the summands in (4.39) are positive, and then
established a positive lower bound for the expectation of the kernel
$\mathfrak{h}$ of $\hat{\sigma}^{2}$ in (4.40); this gives way to using Lemma
4.3 to establish the exponential lower bound in (4.41).

To weaken the condition to the more typical $2m<n$ assumed for the uniform B-E
bound in Theorem 2.1, a possible avenue is to first establish a nonuniform B-E
bound for the _self-normalized U-statistic_

(4.43) $T_{n}^{*}\equiv\frac{\sqrt{n}}{m\hat{\sigma}^{*}}U_{n},$

where
${\hat{\sigma}^{*}}^{2}\equiv\frac{n-1}{(n-m)^{2}}\sum_{i=1}^{n}q_{i}^{2}$,
i.e. establishing a bound of the form

(4.44) $|P(T_{n}^{*}\leq
x)-\Phi(x)|\leq\exp\Bigg{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}+\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{(1+|x|^{3})\sqrt{n}\sigma^{3}}\text{
for }x\in\mathbb{R}$

analogous to (3.2), by employing a similar strategy to how our current B-E
bound for the Studentized $T_{n}$ was established, in which case an
exponential lower bound for ${\sigma^{*}}^{2}$ analogous to (4.41) has to be
established by using Lemma 4.3. In some unreported calculations, we found that
the weaker assumption $2m<n$ suffices to derive the said exponential lower
bound. To leverage (4.44) as a ”bridge” to establish the nonuniform bound for
the Studentized U-statistic $T_{n}$ in (3.2), one can then potentially exploit
the well-known equity of the events

(4.45) $\\{T_{n}>x\\}=\\{T_{n}^{*}>xb_{m,n}(x)\\}\text{ for any }x\geq 0$

that results from the algebraic relationship

(4.46)
$T_{n}=\frac{T_{n}^{*}}{\Big{(}1-\frac{m^{2}(n-1)}{(n-m)^{2}}{T_{n}^{*}}^{2}\Big{)}^{1/2}},$

where we have defined

$b_{m,n}(x)\equiv\bigg{(}1+\frac{m^{2}(n-1)x^{2}}{(n-m)^{2}}\bigg{)}^{-1/2};$

we note in passing that the relationship in (4.46) is analogous to the
relationship between the self-normalized sum $S_{n}/V_{n}$ and $T_{student}$
in (2.14), and has been used in Lai et al. (2011) and Shao and Zhou (2016) to
establish _Cram $\acute{\text{e}}$r-type_ moderate deviation results for
Studentized U-statistics. From (4.44) and (4.45), one can write

(4.47) $|P(T_{n}\leq
x)-{\Phi}(x)|\leq\exp\Bigg{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\Bigg{)}+|\bar{\Phi}(xb_{m,n}(x))+\bar{\Phi}(x)|+\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{(1+|xb_{m,n}(x)|^{3})\sqrt{n}\sigma^{3}},$

and further bound the last two terms on the right; without loss of generality,
we can focus on the range $x\geq 0$. The term
$|\bar{\Phi}(xb_{m,n}(x))+\bar{\Phi}(x)|$ is quite easy to bound, but we skip
the details and refer to Section 5.2.2 for similar arguments used to handle an
analogous quantity for the t-statistic, where we prove Theorem 3.2. However, a
bottleneck arises when attempting to bound the last term in (4.47): Under
$2m<n$ where one has
$b_{m,n}(\sqrt{n})=(1+\frac{m^{2}(n-1)n}{(n-m)^{2}})^{-1/2}\geq(1+\frac{m^{2}n^{2}}{(n-m)^{2}})^{-1/2}\geq(1+4m^{2})^{-1/2}$,
while the nonuniform multiplicative factor in $x$ is seen to be such that

$\frac{1}{1+(xb_{m,n}(x))^{3}}\leq\frac{1}{1+(xb_{m,n}(\sqrt{n}))^{3}}\leq\frac{1}{1+(x(1+4m^{2})^{-1/2})^{3}}\text{
for }0\leq x\leq\sqrt{n},$

the factor doesn’t vanish as $x\rightarrow\infty$ because
$\lim_{x\rightarrow\infty}xb_{m,n}(x)=\frac{n-m}{m\sqrt{n-1}}$. This means,
for the range $x\geq\sqrt{n}$, one has to show that the absolute difference
$|P(T_{n}^{*}>xb_{m,n}(x))-\bar{\Phi}(xb_{m,n}(x))|$ is no larger than our
exponential correction factor
$\exp\big{(}-\frac{c(m)n\sigma^{6}}{(\operatorname{\mathbb{E}}[|h|^{3}])^{2}}\big{)}$,
perhaps up to an absolute multiplicative constant in $m$. We believe this is
possible since both $\bar{\Phi}(xb_{m,n}(x))$ and $P(T_{n}^{*}>xb_{m,n}(x))$
are expected to be small for $x\geq\sqrt{n}$. By a standard upper bound of the
normal survival function $\bar{\Phi}(\cdot)$ (Chen et al., 2011, p.16,
$(2.11)$) and the fact that $xb_{m,n}(x)$ is increasing in $x\geq 0$,

$\displaystyle\bar{\Phi}(xb_{m,n}(x))$
$\displaystyle\leq\min\Bigg{(}\frac{1}{2},\frac{1}{xb_{m,n}(x)\sqrt{2\pi}}\Bigg{)}\exp\Bigg{(}-\frac{(xb_{m,n}(x))^{2}}{2}\Bigg{)}$
$\displaystyle\leq\
\exp\Bigg{(}-\frac{n}{2}\Bigg{(}1+\frac{m^{2}(n-1)n}{(n-m)^{2}}\Bigg{)}^{-1}\Bigg{)}\quad\text{
for }x\geq\sqrt{n},$

which has the desired exponential rate of decay in $n$. Our intuition is that
the term $P(T_{n}^{*}>xb_{m,n}(x))$ is also expected to have some form of
exponential decay in $x$ to induce an exponentially decaying term in $n$, but
important Hoeffding-type inequalities comparable to those available for the
self-normalized sum are missing in the literature; see Lemma 5.1 below. A
quest for such inequalities is an important problem that deserves independent
investigation.

##  5\. Proof of the nonuniform B-E bound for Student’s t-statistic

We now prove the refined nonuniform bound for the Student’s t-statistic and
self-normalized sum in Theorem 3.2. It suffices to consider $x\geq 0$, whether
we are aiming to establish the theorem for the self-normalized sum
$S_{n}/V_{n}$ or the t-statistic $T_{student}$, otherwise one can replace the
$X_{i}$’s with $-X_{i}$’s instead. A technical tool that we will use is the
following Hoeffding-type bound that can be found in de la Peña et al. (2009,
Theorem 2.16, p.12):

######  Lemma 5.1 (Sub-Gaussian property for self-normalized sums).

Under the same assumptions as Theorem 3.2, it is true that, for any $x\geq 0$,

$P\Big{(}S_{n}>x(4\sqrt{n}\|X_{1}\|_{2}+V_{n})\Big{)}\leq 2e^{-x^{2}/2}.$

###  5.1. Proof for the self-normalized sum

We will first prove a more general bound for the self-normalized sum:

(5.1)
$|P(S_{n}/V_{n}>x)-\bar{\Phi}(x)|\leq\begin{cases}&C\frac{(1+x)^{2}}{e^{x^{2}/2}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\text{
for }0\leq x\leq n^{1/6}\frac{\|X_{1}\|_{2}}{\|X_{1}\|_{3}};\\\
&\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+2\exp\Bigg{(}-\frac{x^{2}}{162}\Bigg{)}\text{
for }x>n^{1/6}\frac{\|X_{1}\|_{2}}{\|X_{1}\|_{3}}.\end{cases}$

Now we prove (5.1). As a simple consequence of _Cram $\acute{\text{e}}$r-type
moderate deviation_ for self-normalized sums (Jing et al., 2003, Theorem 2.3),
one can derive the nonuniform B-E bound

(5.2) $|P(S_{n}/V_{n}>x)-\bar{\Phi}(x)|\leq
C\frac{(1+x)^{2}}{e^{x^{2}/2}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\text{
for }0\leq x\leq n^{1/6}\frac{\|X_{1}\|_{2}}{\|X_{1}\|_{3}};$

see Jing et al. (2003, Eqn. $(2.11)$, p.2171). For any
$x>n^{1/6}\|X_{1}\|_{2}/\|X_{1}\|_{3}$,

$\displaystyle P\bigg{(}S_{n}/V_{n}>x\bigg{)}$ $\displaystyle\leq
P\bigg{(}V_{n}\leq\sqrt{n}\|X_{1}\|_{2}/2\bigg{)}+P(S_{n}>xV_{n},V_{n}>\sqrt{n}\|X_{1}\|_{2}/2)$
$\displaystyle\leq
P\Bigg{(}\frac{V_{n}^{2}}{n}\leq\frac{\operatorname{\mathbb{E}}[X_{1}^{2}]}{4}\Bigg{)}+P\Bigg{(}S_{n}>\frac{x(4\sqrt{n}\|X_{1}\|_{2}+V_{n})}{9}\Bigg{)}$
$\displaystyle\leq\exp\Bigg{(}\frac{-\frac{n}{2}(\frac{3}{4}\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{1.5(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+2\exp\Bigg{(}-\frac{x^{2}}{162}\Bigg{)}\text{
by Lemmas~{}\ref{lem:chernoff_lower_tail_bdd_U_stat} and
\ref{lem:subGauss_sn_sum}}$ (5.3)
$\displaystyle\leq\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+2\exp\Bigg{(}-\frac{x^{2}}{162}\Bigg{)}.$

Moreover, by the standard normal tail bound,

(5.4) $\bar{\Phi}(x)\leq\frac{1}{2}e^{-x^{2}/2}\leq
2\exp\Bigg{(}-\frac{x^{2}}{162}\Bigg{)}$

Combining (5.2) for $0\leq x\leq n^{1/6}\|X_{1}\|_{2}/\|X_{1}\|_{3}$ along
with (5.3) and (5.4) for $x>n^{1/6}\|X_{1}\|_{2}/\|X_{1}\|_{3}$, we get that
the bound (5.1) for the self-normalized sum.

Lastly, the term $2e^{-x^{2}/162}$ in the bound from (5.1) can be bounded as

(5.5)
$2e^{-x^{2}/162}=2e^{-x^{2}/324}\underbrace{\exp\left(-\frac{x^{2}}{324}+3\log(x)\right)}_{\leq
C}x^{-3}\\\ \leq
Ce^{-cx^{2}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\text{
for }x>n^{1/6}\frac{\|X_{1}\|_{2}}{\|X_{1}\|_{3}}.$

Combining (5.1) and (5.5) and suitably adjusting the absolute constants, we
get the desired bound

(5.6)
$|P(S_{n}/V_{n}>x)-\bar{\Phi}(x)|\leq\frac{C}{e^{cx^{2}}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}+\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}\text{
for all }x\geq 0.$

(In particular, this means, for the self-normalized sum, the constant $C_{1}$
in Theorem 3.2 can be simply taken to be $1$.)

###  5.2. Proof for the t-statistic

To prove the theorem for $T_{student}$, we will adapt a ”bridging” argument
found in Wang and Jing (1999): Define the function

$a_{n}(x)=a_{n,x}\equiv\Bigg{(}\frac{n}{n+x^{2}-1}\Bigg{)}^{1/2},$

which has the property that

(5.7) $1/\sqrt{2}\leq a_{n,x}\leq\sqrt{2}\text{ for }0\leq x\leq\sqrt{n},$

considering that $n\geq 2$. Moreover, the function $xa_{n}(x)$ is increasing
in $x$ because

(5.8)
$\frac{d}{dx}xa_{n}(x)=\bigg{(}\frac{n}{n+x^{2}-1}\bigg{)}^{1/2}\bigg{(}1-\frac{x^{2}}{n+x^{2}-1}\bigg{)}>0$

for $n\geq 2$. Using the well-known algebraic relationship in (2.14), we have
the event equivalence

$\Big{\\{}T_{student}>x\Big{\\}}=\Bigg{\\{}\frac{S_{n}}{V_{n}}>xa_{n}(x)\Bigg{\\}}\text{
for any }x\geq 0.$

Then, by the triangular inequality we have

(5.9)
$|P(T_{student}>x)-\bar{\Phi}(x)|\leq|P(S_{n}/V_{n}>xa_{n}(x))-\bar{\Phi}(xa_{n}(x))|+|\bar{\Phi}(xa_{n}(x))-\bar{\Phi}(x)|.$

####  5.2.1. Bounding $|P(S_{n}/V_{n}>xa_{n}(x))-\overline{\Phi}(xa_{n}(x))|$

From (5.7), it must be true that for any small constant $c>0$,

$\frac{(1+xa_{n,x})^{2}}{e^{c(xa_{n,x})^{2}}}\leq\frac{(1+\sqrt{2}x)^{2}}{e^{cx^{2}/2}}\text{
for }0\leq x\leq\sqrt{n},$

which also implies

(5.10)
$\big{|}P(S_{n}/V_{n}>xa_{n,x})-\bar{\Phi}(xa_{n,x})\big{|}\leq\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+C\frac{(1+x)^{2}}{e^{cx^{2}}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\\\
\text{ for }0\leq x\leq\sqrt{n}$

from (5.6) by adjusting the constants $C,c$. Since $xa_{n,x}$ is increasing by
(5.8),

(5.11) $xa_{n,x}\geq\sqrt{n}a_{n}(\sqrt{n})=\frac{n}{\sqrt{2n-1}}\text{ for
}x\geq\sqrt{n}.$

As $n/\sqrt{2n-1}\geq n^{1/6}\|X_{1}\|_{2}/\|X_{1}\|_{3}$, we can then apply
(5.1) to get

$\displaystyle\big{|}P(S_{n}/V_{n}>xa_{n,x})-\bar{\Phi}(xa_{n,x})\big{|}$
$\displaystyle\leq\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+2\exp\Bigg{(}-\frac{x^{2}a_{n,x}^{2}}{162}\Bigg{)}$
(5.12)
$\displaystyle\leq\exp\Bigg{(}\frac{-n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{16(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+2e^{-162^{-1}n^{2}/(2n-1)}\text{
for }x\geq\sqrt{n},\text{ by }\eqref{something}$

Combining (5.10) and (5.12), as well as
$(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}/(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}\leq
1$, upon adjusting the absolute constants we have

(5.13) $\big{|}P(S_{n}/V_{n}>xa_{n,x})-\bar{\Phi}(xa_{n,x})\big{|}\leq
C_{1}\exp\Bigg{(}\frac{-c_{1}n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+\frac{C_{2}}{e^{c_{2}x^{2}}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\\\
\text{ for all }x\geq 0.$

####  5.2.2. Bounding $|\overline{\Phi}(xa_{n}(x))-\overline{\Phi}(x)|$

First write the inequality

$|xa_{n,x}-x|=\left|\frac{(a_{n,x}^{2}-1)x}{a_{n,x}+1}\right|=\left|\left(\frac{1-x^{2}}{n+x^{2}-1}\right)\left(\frac{x}{a_{n,x}+1}\right)\right|\leq\frac{(1+x^{2})x}{(n-1)(a_{n,x}+1)};$

the prior inequality in turns implies, via Taylor’s theorem,

(5.14) $|\Phi(xa_{n,x})-\Phi(x)|\leq\phi\Big{(}x(a_{n,x}\wedge
1)\Big{)}\Bigl{|}xa_{n,x}-x\Bigr{|}\\\ \leq\phi\Big{(}x(a_{n,x}\wedge
1)\Big{)}\frac{(1+x^{2})x}{(n-1)(a_{n,x}+1)}=\frac{(1+x^{2})x}{\sqrt{2\pi}(n-1)(a_{n,x}+1)}\exp\left(\frac{-x^{2}(a_{n,x}\wedge
1)^{2}}{2}\right).$

by the mean-value theorem. At the same time, we also have

(5.15)
$|\Phi(xa_{n,x})-\Phi(x)|\leq\bar{\Phi}(xa_{n,x})+\bar{\Phi}(x)\leq\exp\left(\frac{-x^{2}(a_{n,x}\wedge
1)^{2}}{2}\right)$

by the typical normal tail bound; see Chen et al. (2011, Eqn. (2.11)) for
instance. Combining (5.14) and (5.15), we have

(5.16) $|\Phi(xa_{n,x})-\Phi(x)|\leq\\\
\min\left(\frac{(1+x^{2})x}{\sqrt{2\pi}(n-1)(a_{n,x}+1)},1\right)\exp\left(\frac{-x^{2}(a_{n,x}\wedge
1)^{2}}{2}\right).$

Now, for the range $0\leq x\leq n^{1/6}$, from (5.16) and (5.7) one get

(5.17)
$|\Phi(xa_{n,x})-\Phi(x)|\leq\frac{(1+n^{1/3})n^{1/6}}{\sqrt{\pi}(1+\sqrt{2})(n-1)}\exp\Big{(}\frac{-x^{2}}{4}\Big{)}\leq\frac{C}{\sqrt{n}}\exp\Big{(}\frac{-x^{2}}{4}\Big{)}\text{
for }0\leq x\leq n^{1/6}.$

For the range $n^{1/6}<x\leq n^{1/2}$, since $a_{n,n^{1/6}}\leq 1$, from
(5.16) one get

$\displaystyle|\Phi(xa_{n,x})-\Phi(x)|$
$\displaystyle\leq\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{4}\Big{)}\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{4}\Big{)}$
$\displaystyle=\underbrace{\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{4}+3\log(xa_{n,x})\Big{)}}_{\leq
C}\frac{1}{x^{3}a_{n,x}^{3}}\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{4}\Big{)}$
$\displaystyle\leq\frac{C}{n^{1/2}a_{n,x}^{3}}\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{4}\Big{)}$
$\displaystyle\leq\frac{C}{n^{1/2}a_{n,n^{1/2}}^{3}}\exp\Big{(}-\frac{x^{2}a_{n,n^{1/2}}^{2}}{4}\Big{)}$
(5.18)
$\displaystyle\leq\frac{2^{3/2}C}{\sqrt{n}}\exp\Big{(}-\frac{x^{2}}{8}\Big{)}\text{
for }n^{1/6}<x\leq n^{1/2},$

where the last inequality uses (5.7). For the range $x>n^{1/2}$, using that
$xa_{n,x}$ is increasing in $x$ from (5.8) again, from (5.16) we get that

(5.19)
$|\Phi(xa_{n,x})-\Phi(x)|\leq\exp\Big{(}-\frac{x^{2}a_{n,x}^{2}}{2}\Big{)}\leq\exp\Big{(}-\frac{na_{n,n^{1/2}}^{2}}{2}\Big{)}\leq\exp\big{(}-n/4\big{)}\text{
for }x>n^{1/2},$

where the last inequality again uses (5.7). Combining (5.17), (5.18) and
(5.19), we get

(5.20)
$|\Phi(xa_{n,x})-\Phi(x)|\leq\exp\Big{(}-\frac{n}{4}\Big{)}+\frac{C}{\sqrt{n}}\exp\Big{(}-\frac{x^{2}}{8}\Big{)}\text{
for }x\geq 0.$

Lastly, combining (5.9), (5.13) and (5.20), we get

$|P(T_{student}>x)-\bar{\Phi}(x)|\leq
C_{1}\exp\Bigg{(}\frac{-c_{1}n(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3}}{(\operatorname{\mathbb{E}}[X_{1}^{3}])^{2}}\Bigg{)}+\frac{C_{2}}{e^{c_{2}x^{2}}}\frac{\operatorname{\mathbb{E}}[|X_{1}|^{3}]}{\sqrt{n}(\operatorname{\mathbb{E}}[X_{1}^{2}])^{3/2}}\text{
for all }x\geq 0$

because $\|X_{1}\|_{3}/\|X_{1}\|_{2}\geq 1$, and Theorem 3.2 for $T_{student}$
is proved.

## Acknowledgments

This research is partially supported by National Nature Science Foundation of
China NSFC 12031005 and Shenzhen Outstanding Talents Training Fund, China.

## Appendices

These appendices are organized as follows: Appendix A first list and prove
some supporting lemmas, with the remaining appendices covering the proofs for:

  * •

Appendix B: Lemmas 4.1

  * •

Appendix C: Lemma 4.2

  * •

Appendix D: Lemmas 4.3 and 4.4

##  Appendix A Technical lemmas

This appendix lists out a few sets of useful results that, for the most part,
have already been established in our related work Leung and Shao (2023),
except for Lemma A.8. For any $x\in\mathbb{R}$, recall that the Stein equation
(Stein, 1972)

(A.1) $f^{\prime}(w)-wf(w)=I(w\leq x)-\Phi(x),$

has the unique bounded solution $f(w)=f_{x}(w)$ of the form

(A.2)
$f_{x}(w)\equiv\begin{cases}\sqrt{2\pi}e^{w^{2}/2}\Phi(w)\bar{\Phi}(x)&w\leq
x;\\\ \sqrt{2\pi}e^{w^{2}/2}\Phi(x)\bar{\Phi}(w)&w>x;\end{cases}$

see Chen et al. (2011, p.14). Since $f_{x}$ as in (A.2) is not differentiable
at $w=x$, we customarily define

(A.3) $f_{x}^{\prime}(x)\equiv xf_{x}(x)+1-\Phi(x),$

so (A.1) holds for all $w\in\mathbb{R}$.

###  A.1. Properties of the solution to Stein’s equation

This section provides some useful bounds related to $f_{x}$ in (A.2). We will
define

(A.4) $g_{x}(w)\equiv(wf_{x}(w))^{\prime}=f_{x}(w)+wf_{x}^{\prime}(w),$

where it is understood that $g_{x}(x)\equiv f_{x}(x)+xf_{x}^{\prime}(x)$ for
$f_{x}^{\prime}(x)$ defined in (A.3). Precisely,

(A.5)
$f_{x}^{\prime}(w)=\begin{cases}\left(\sqrt{2\pi}we^{w^{2}/2}\Phi(w)+1\right)\bar{\Phi}(x)&\text{
for }\quad w\leq x;\\\
\left(\sqrt{2\pi}we^{w^{2}/2}\bar{\Phi}(w)-1\right)\Phi(x)&\text{ for }\quad
w>x;\end{cases}$ (A.6)
$g_{x}(w)=\begin{cases}\sqrt{2\pi}\bar{\Phi}(x)\left((1+w^{2})e^{w^{2}/2}\Phi(w)+\frac{w}{\sqrt{2\pi}}\right)&\text{
for }\quad w\leq x;\\\
\sqrt{2\pi}\Phi(x)\left((1+w^{2})e^{w^{2}/2}\bar{\Phi}(w)-\frac{w}{\sqrt{2\pi}}\right)&\text{
for }\quad w>x.\end{cases}$

######  Lemma A.1 (Uniform bounds).

For $f_{x}$ and $f_{x}^{\prime}$, the following bounds are true:

$|f_{x}^{\prime}(w)|\leq 1,\quad\quad 0<f_{x}(w)\leq 0.63\quad\text{ and
}\quad 0\leq g_{x}(w)\quad\text{ for all }\quad
w,x\in\operatorname{\mathbb{R}}.$

Moreover, for any $x\in[0,1]$, $g_{x}(w)\leq 2.3$ for all
$w\in\operatorname{\mathbb{R}}$.

######  Lemma A.2 (Nonuniform bounds when $x\geq 1$).

For $x\geq 1$, the following are true:

(A.7) $f_{x}(w)\leq\begin{cases}1.7e^{-x}&\text{ for }\quad w\leq x-1;\\\
1/x&\text{ for }\quad x-1<w\leq x;\\\ 1/w&\text{ for }\quad x<w;\end{cases}$

and

(A.8) $|f_{x}^{\prime}(w)|\leq\begin{cases}e^{1/2-x}&\text{ for }\quad w\leq
x-1;\\\ 1&\text{ for }\quad x-1<w\leq x;\\\ (1+x^{2})^{-1}&\text{ for }\quad
w>x.\end{cases}$

Moreover, $g_{x}(w)\geq 0$ for all $w\in\mathbb{R}$,

(A.9) $g_{x}(w)\leq\begin{cases}1.6\;\bar{\Phi}(x)&\text{ for }\quad w\leq
0;\\\ 1/w&\text{ for }\quad w>x,\end{cases}$

and $g_{x}(w)$ is increasing for $0\leq w\leq x$ with

$g_{x}(x-1)\leq xe^{1/2-x}\;\ \text{ and }\;\ g_{x}(x)\leq x+2.$

######  Lemma A.3 (Bound on expectation of $f_{x}(W_{b})$ when $x\geq 1$).

Let $\xi_{1},\dots,\xi_{n}$ be independent random variables with
$\operatorname{\mathbb{E}}[\xi_{i}]=0$ for all $i=1,\dots,n$ and
$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]\leq 1$, and define
$\xi_{b,i}=\xi_{i}I(|\xi_{i}|\leq 1)+1I(\xi_{i}>1)-1I(\xi_{i}<-1)$ and
$W_{b}=\sum_{i=1}^{n}\xi_{b,i}$. For $x\geq 1$, then there exists an absolute
constant $C>0$ such that

$|\operatorname{\mathbb{E}}[f_{x}(W_{b})]|\leq Ce^{-x}.$

###### Proof of Lemma A.3.

From (A.7) in Lemma A.2 and $|f_{x}|\leq 0.63$ in Lemma A.1,

$\displaystyle|\operatorname{\mathbb{E}}[f_{x}(W_{b})]|$ $\displaystyle\leq
1.7e^{-x}+|\operatorname{\mathbb{E}}[f_{x}(W_{b})I(W_{b}>x-1)]|\leq
1.7e^{-x}+e^{1-x}0.63\operatorname{\mathbb{E}}[e^{W_{b}}],$

then apply the Bennett inequality in Lemma A.5 below. ∎

###  A.2. Bounds for the censored summands $\xi_{b,i}$’s and their sum $W_{b}$

The following bounds for the censored summands $\xi_{b,i}$’s and their sum
$W_{b}$ will be useful.

######  Lemma A.4 (Bound on expectation of $\xi_{b,i}$).

Let $\xi_{b,i}=\xi_{i}I(|\xi_{i}|\leq 1)+1I(\xi_{i}>1)-1I(\xi_{i}<-1)$ with
$\operatorname{\mathbb{E}}[\xi_{i}]=0$. Then

$\left|\operatorname{\mathbb{E}}[\xi_{b,i}]\right|\leq\operatorname{\mathbb{E}}[\xi_{i}^{2}I(|\xi_{i}|>1)]\leq\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]\wedge\operatorname{\mathbb{E}}[\xi_{i}^{2}]$

######  Lemma A.5 (Bennett’s inequality for a sum of censored random
variables).

Let $\xi_{1},\dots,\xi_{n}$ be independent random variables with
$\operatorname{\mathbb{E}}[\xi_{i}]=0$ for all $i=1,\dots,n$ and
$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]\leq 1$, and define
$\xi_{b,i}=\xi_{i}I(|\xi_{i}|\leq 1)+1I(\xi_{i}>1)-1I(\xi_{i}<-1)$. For any
$t>0$ and $W_{b}=\sum_{i=1}^{n}\xi_{b,i}$, we have

$\operatorname{\mathbb{E}}[e^{tW_{b}}]\leq\exp\left(e^{2t}/4-1/4+t/2\right)$

######  Lemma A.6 (Exponential randomized concentration inequality for a sum
of censored random variables).

Let $\xi_{1},\dots,\xi_{n}$ be independent random variables with mean zero and
finite second moments, and for each $i=1,\dots,n$, define

$\xi_{b,i}=\xi_{i}I(|\xi_{i}|\leq 1)+1I(\xi_{i}>1)-1I(\xi_{i}<-1),$

an upper-and-lower censored version of $\xi_{i}$; moreover, let
$W=\sum_{i=1}^{n}\xi_{i}$ and $W_{b}=\sum_{i=1}^{n}\xi_{b,i}$ be their
corresponding sums, and $\Delta_{1}$ and $\Delta_{2}$ be two random variables
on the same probability space. Assume there exists $c_{1}>c_{2}>0$ and
$\delta\in(0,1/2)$ such that

$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]\leq c_{1}$

and

$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|\min(\delta,|\xi_{i}|/2)]\geq
c_{2}.$

Then for any $\lambda\geq 0$, it is true that

$\displaystyle\operatorname{\mathbb{E}}[e^{\lambda W_{b}}I(\Delta_{1}\leq
W_{b}\leq\Delta_{2})]$ $\displaystyle\leq\left(\mathbb{E}\left[e^{2\lambda
W_{b}}\right]\right)^{1/2}\exp\left(-\frac{c_{2}^{2}}{16c_{1}\delta^{2}}\right)$
$\displaystyle+\frac{2e^{\lambda\delta}}{c_{2}}\Biggl{\\{}2\sum_{i=1}^{n}\mathbb{E}[|\xi_{b,i}|e^{\lambda
W_{b}^{(i)}}(|\Delta_{1}-\Delta_{1}^{(i)}|+|\Delta_{2}-\Delta_{2}^{(i)}|)]$
$\displaystyle+\mathbb{E}[|W_{b}|e^{\lambda
W_{b}}(|\Delta_{2}-\Delta_{1}|+2\delta)]$
$\displaystyle+\sum_{i=1}^{n}\Big{|}\operatorname{\mathbb{E}}[\xi_{b,i}]\Big{|}\operatorname{\mathbb{E}}[e^{\lambda
W_{b}^{(i)}}(|\Delta_{2}^{(i)}-\Delta_{1}^{(i)}|+2\delta)]\Biggr{\\}},$

where $\Delta_{1}^{(i)}$ and $\Delta_{2}^{(i)}$ are any random variables on
the same probability space such that $\xi_{i}$ and
$(\Delta_{1}^{(i)},\Delta_{2}^{(i)},W^{(i)},W_{b}^{(i)})$ are independent,
where $W^{(i)}=W-\xi_{i}$ and $W_{b}^{(i)}=W_{b}-\xi_{b,i}$. In particular, if
$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]=1$, one can take

$\delta=\frac{\beta_{2}+\beta_{3}}{4},\quad\lambda=\frac{1}{2},\quad
c_{1}=1,\quad c_{2}=\frac{1}{4},$

where
$\beta_{2}\equiv\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}I(|\xi_{i}|>1)]$
and
$\beta_{3}\equiv\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{3}I(|\xi_{i}|\leq
1)]$.

###  A.3. Bounds related to the components of the censored denominator
remainder in Section 4

This subsection supplements Section 4, and in particular (1.2) and (4.2) are
assumed to hold. Given

$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]+\sum_{i=1}^{n}\operatorname{\mathbb{E}}[(\xi_{i}^{2}-1)I(|\xi_{i}|>1)]=\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]=1$

and $d_{n}^{2}=n/(n-1)$, we shall rewrite
$D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}$ in (4.22) as

(A.10)
$D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}=d_{n}^{2}\Bigg{(}\delta_{0,b}+\bar{\delta}_{1}+\bar{\delta}_{2,b}\Bigg{)}+\frac{\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]}{n-1}-\sum_{i=1}^{n}\operatorname{\mathbb{E}}\Big{[}(\xi_{i}^{2}-1)I(|\xi_{i}|>1)\Big{]},$

where

$\delta_{0,b}\equiv\sum_{i=1}^{n}\Big{(}\xi_{b,i}^{2}-\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]\Big{)}.$

This section includes some useful properties related to the components
$\delta_{0,b}$, $\delta_{1}$ and $\delta_{2,b}$ in (A.10); recall $f_{x}$ is
the solution to the Stein equation in (A.2).

######  Lemma A.7 (Properties of $\delta_{0,b}$).

There exists positive absolute constants $C>0$ and $x\geq 1$,

(A.11)
$\operatorname{\mathbb{E}}[\delta_{0,b}^{2}]\leq\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{b,i}|^{3}]$
(A.12) $\operatorname{\mathbb{E}}\Big{[}e^{W_{b}}\delta_{0,b}^{2}\Big{]}\leq
C\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{b,i}|^{3}]$

and

(A.13) $\Big{|}\operatorname{\mathbb{E}}[\delta_{0,b}f_{x}(W_{b})]\Big{|}\leq
Ce^{-x}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{b,i}|^{3}].$

###### Proof of Lemma A.7.

The proofs of (A.11) and (A.12) can be found in Leung and Shao (2023, Appendix
D.1, p.30-31); the proof of (A.13) can be found in Leung and Shao (2023,
Appendix D.2, p.33). Note that $\delta_{0,b}$ is same as the quantity
”$\Pi_{1}$” in Leung and Shao (2023). ∎

######  Lemma A.8 (Properties of $\delta_{1}$).

Assume $\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})]=0$ and $\sigma^{2}=1$.
For some positive absolute constants $C(m)>0$,

(A.14)
$\operatorname{\mathbb{E}}\Big{[}|\delta_{1}|\Big{]}\leq\frac{Cm^{2}\operatorname{\mathbb{E}}[h^{2}]}{n}$

and

(A.15) $\operatorname{\mathbb{E}}\Big{[}|\delta_{1}|^{3/2}\Big{]}\leq
C(m)\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}.$

###### Proof of Lemma A.8.

In Leung and Shao (2023, Section 3, p.11), we have established that

$\operatorname{\mathbb{E}}[|\delta_{1}^{*}|]\leq
2\left[\frac{m(m-1)(n-1)}{(n-m)^{2}}\right]+\frac{4(n-1)^{2}(m-1)^{2}}{(n-m)^{2}(n-m+1)m}\operatorname{\mathbb{E}}\left[h^{2}\right].$

Moreover, by Serfling (1980, Lemma 5.2.1.A$(i)$, p.183),
$\operatorname{\mathbb{E}}[U_{n}^{2}]\leq\frac{m\operatorname{\mathbb{E}}[h^{2}]}{n}$.
Collecting these facts give (A.14) because
$\operatorname{\mathbb{E}}[|h|^{2}]\geq 1$.

For (A.15), we need to first establish certain higher moment bounds for
$W_{n}$ and $\Lambda_{n}$. For $W_{n}$, by Rosenthal (1970, Theorem 3)’s
inequality, we have that

(A.16) $\operatorname{\mathbb{E}}\Big{[}|W_{n}|^{3}\Big{]}\leq
C\Bigg{\\{}\Big{(}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}]\Big{)}^{3/2}+\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]\Bigg{\\}}\leq
C\Bigg{(}1+\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}\Bigg{)}.$

For $\Lambda_{n}$, upon rescaling $\Lambda_{n}^{2}$ with the factor
$n{n\choose 2m-1}^{-1}$, we write:

$\displaystyle\tilde{\Lambda}_{n}^{2}\equiv n{n\choose
2m-1}^{-1}\Lambda_{n}^{2}$ $\displaystyle={n\choose
2m-1}^{-1}\sum_{i=1}^{n}\left(\sum_{\begin{subarray}{c}1\leq
i_{1}<\cdots<i_{m-1}\leq n\\\ i_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}\bar{h}_{m}(X_{i},X_{i_{1}},\dots,X_{i_{m-1}})\right)^{2}$
$\displaystyle={n\choose
2m-1}^{-1}\sum_{i=1}^{n}\sum_{\begin{subarray}{c}1\leq
i_{1}<\cdots<i_{m-1}\leq n\\\ 1\leq j_{1}<\cdots<j_{m-1}\leq n\\\
i_{l},j_{l}\neq i\text{ for
}l=1,\dots,m-1\end{subarray}}\bar{h}_{m}(X_{i},X_{i_{1}},\dots,X_{i_{m-1}})\bar{h}_{m}(X_{i},X_{j_{1}},\dots,X_{j_{m-1}})$
(A.17) $\displaystyle={n\choose 2m-1}^{-1}\left(\sum_{k=m}^{2m-1}\sum_{1\leq
i_{1}<\dots<i_{k}\leq
n}\bar{\operatorname{\mathcal{H}}}_{k}(X_{i_{1}},\dots,X_{i_{k}})\right),$

where for each $k=m,\dots,2m-1$,
$\bar{\operatorname{\mathcal{H}}}_{k}:\mathbb{R}^{k}\longrightarrow\mathbb{R}$
is a symmetric kernel of degree $k$ defined as

(A.18)
$\bar{\operatorname{\mathcal{H}}}_{k}(x_{1},\dots,x_{k})\equiv(2m-k)\times\sum_{\begin{subarray}{c}\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,\dots,k\\}:\\\
|\operatorname{\mathcal{S}}_{1}|=2m-k\\\
|\operatorname{\mathcal{S}}_{2}|=|\operatorname{\mathcal{S}}_{3}|=k-m\\\
\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\text{disjoint}\end{subarray}}\bar{h}_{m}(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(x_{\operatorname{\mathcal{S}}_{1}},x_{\operatorname{\mathcal{S}}_{3}})$

induced by $h(\cdot)$. Hence, up to scaling factors, $\Lambda_{n}^{2}$ can be
seen as a sum of $2m-1$ U-statistics with kernels of degree $k=m,\dots,2m-1$.
Moreover, for $k=2m-1$,
$\bar{\operatorname{\mathcal{H}}}_{2m-1}(x_{1},\dots,x_{m})$ is seen to have
the _second-order_ degeneracy property

(A.19)
$\operatorname{\mathbb{E}}[\bar{\operatorname{\mathcal{H}}}_{2m-1}(X_{1},\dots,X_{2m-1})|X_{i},X_{j}]=0\text{
for }\\{i,j\\}\subset\\{1,\dots,2m-1\\}\text{ and }i\neq j,$

which, in particular, implies

(A.20)
$\operatorname{\mathbb{E}}[\bar{\operatorname{\mathcal{H}}}_{2m-1}(X_{1},\dots,X_{2m-1})]=0;$

we will prove (A.19) at the end.

Now, by taking the $(3/2)$-th absolute central moment of
$\tilde{\Lambda}_{n}^{2}$, from (A.17) one get

$\displaystyle\operatorname{\mathbb{E}}\Bigg{[}\Bigg{|}\tilde{\Lambda}_{n}^{2}-\operatorname{\mathbb{E}}[\tilde{\Lambda}_{n}^{2}]\Bigg{|}^{3/2}\Bigg{]}$
$\displaystyle\leq
C(m)\sum_{k=m}^{2m-1}\frac{1}{n^{3(2m-1-k)/2}}\operatorname{\mathbb{E}}\Bigg{[}\Bigg{|}\frac{\sum_{1\leq
i_{1}<\dots<i_{k}\leq
n}\bar{\operatorname{\mathcal{H}}}_{k}(X_{i_{1}},\dots,X_{i_{k}})}{{n\choose
k}}-\operatorname{\mathbb{E}}[\bar{\operatorname{\mathcal{H}}}_{k}]\Bigg{|}^{3/2}\Bigg{]}$
$\displaystyle\leq
C(m)\Bigg{\\{}n^{-3/2}\operatorname{\mathbb{E}}\Big{[}|\bar{\operatorname{\mathcal{H}}}_{2m-1}(X_{1},\dots,X_{k})|^{3/2}\Big{]}+\sum_{k=m}^{2m-2}\underbrace{\frac{1}{n^{3m-1-3k/2}}}_{\leq
n^{-2}}\operatorname{\mathbb{E}}\Big{[}|\bar{\operatorname{\mathcal{H}}}_{k}(X_{1},\dots,X_{k})|^{3/2}\Big{]}\Bigg{\\}}$
(A.21) $\displaystyle\leq
C(m)\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}$

where the second last inequality uses the moment bound (4.42) for centered
U-statistics in Lemma 4.4 and the degeneracy of
$\bar{\operatorname{\mathcal{H}}}_{2m-1}$ in (A.19); the last inequality in
(A.21) uses the definition in (A.18), the Cauchy inequality

$\operatorname{\mathbb{E}}[|\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|^{3/2}]\leq\operatorname{\mathbb{E}}[|\bar{h}_{m}(X_{1},\dots,X_{m})|^{3}]$

and that $\operatorname{\mathbb{E}}[|\bar{h}_{m}|^{3}]\leq
C(m)\operatorname{\mathbb{E}}[|h_{m}|^{3}]$ from (4.11). On the other hand,
from the definition in (A.18) and that
$\operatorname{\mathbb{E}}[\bar{h}_{m}^{2}]\leq
C(m)\operatorname{\mathbb{E}}[h_{m}^{2}]$ from (4.11) 444Actually it can also
be shown that
$\operatorname{\mathbb{E}}[\bar{h}_{m}^{2}]\leq\operatorname{\mathbb{E}}[h_{m}^{2}]$;
see $(10.76)$ in Chen et al. (2011, p.284)., it follows that

$\Big{|}\operatorname{\mathbb{E}}[\bar{\operatorname{\mathcal{H}}}_{k}(X_{1},\dots,X_{k})]\Big{|}\leq
C(m)\operatorname{\mathbb{E}}[h^{2}(X_{1},\dots,X_{m})]\text{ for
}k=1,\dots,2m-2,$

which, together with (A.20), implies that

(A.22) $\operatorname{\mathbb{E}}[\tilde{\Lambda}_{n}^{2}]\leq
C(m)\frac{\operatorname{\mathbb{E}}[h^{2}]}{n}$

Combining (A.21) and (A.22), by
$\|h(X_{1},\dots,X_{m})\|_{2}\leq\|h(X_{1},\dots,X_{m})\|_{3}$, we have

(A.23)
$\operatorname{\mathbb{E}}\Big{[}|\tilde{\Lambda}_{n}|^{3/2}\Big{]}=\operatorname{\mathbb{E}}\Bigg{[}\Bigg{(}n{n\choose
2m-1}^{-1}\Bigg{)}^{3/2}|\Lambda_{n}|^{3}\Bigg{]}\leq\frac{C(m)\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}$

Now we can finish proving the lemma, by using the basic property that

(A.24) $\frac{2(n-1)(m-1)}{(n-m)^{2}{n-1\choose
m-1}}\Bigg{|}\sum_{i=1}^{n}W_{n}\Psi_{n,i}\Bigg{|}\leq\frac{n(m-1)^{2}}{(n-m)^{2}}W_{n}^{2}+\frac{(n-1)^{2}}{{n-1\choose
m-1}^{2}(n-m)^{2}}\Lambda_{n}^{2},$

a consequence of the Cauchy’s inequality $2|W_{n}\sum_{i=1}^{n}\Psi_{n,i}|\leq
2\sqrt{n}|W_{n}|\Lambda_{n}$. Recalling the definition of $\delta_{1}^{*}$ in
(4.14), we can apply (A.16) and (A.23) to get

$\displaystyle\operatorname{\mathbb{E}}[|\delta_{1}^{*}|^{3/2}]$
$\displaystyle\leq
C\Bigg{\\{}\left[\frac{n(m-1)^{2}}{(n-m)^{2}}+\frac{2(m-1)}{(n-m)}\right]^{3/2}\operatorname{\mathbb{E}}[|W_{n}|^{3}]+\Bigg{[}\frac{(n-1)^{2}}{{n-1\choose
m-1}^{2}(n-m)^{2}}\Bigg{]}^{3/2}\operatorname{\mathbb{E}}[|\Lambda_{n}|^{3}]\Bigg{\\}}$
$\displaystyle\leq
C(m)\Bigg{\\{}n^{-3/2}\Bigg{(}1+\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}\Bigg{)}+\Bigg{[}\frac{(n-1)^{2}{n\choose
2m-1}}{n{n-1\choose
m-1}^{2}(n-m)^{2}}\Bigg{]}^{3/2}\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}\Bigg{\\}}$
$\displaystyle\leq C(m)\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}}.$

Since $\operatorname{\mathbb{E}}[|U_{n}|^{3}]\leq
C(m)n^{-3/2}\operatorname{\mathbb{E}}[|h|^{3}]$ by the bound (4.42) in Lemma
4.4, we get from (4.16) that

$\operatorname{\mathbb{E}}[|\delta_{1}|^{3/2}]\leq
C(m)\left(\operatorname{\mathbb{E}}[|\delta_{1}^{*}|^{3/2}]+\operatorname{\mathbb{E}}[|U_{n}|^{3}]\right)\leq
C(m)\frac{\operatorname{\mathbb{E}}[|h|^{3}]}{n^{3/2}};$

(A.15) is proved.

It remains to show (A.19), for which we will leverage the degenerate property
of $\bar{h}_{m}$ in (4.9). From the definition in (A.18), it suffices to show
that each summand of
$\bar{\operatorname{\mathcal{H}}}_{2m-1}(X_{1},\dots,X_{2m-1})$ has the same
property, i.e.

$\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{i},X_{j}]=0$

for any disjoint
$\operatorname{\mathcal{S}}_{1},\operatorname{\mathcal{S}}_{2},\operatorname{\mathcal{S}}_{3}\subset\\{1,,\dots,2m-1\\}$
such that $|\operatorname{\mathcal{S}}_{1}|=1$. We consider three cases
555It’s possible that $i\in\operatorname{\mathcal{S}}_{3}$,
$j\in\operatorname{\mathcal{S}}_{2}$ for case $(ii)$ and
$i,j\in\operatorname{\mathcal{S}}_{3}$ for case $(iii)$, with the same proof.:

  1. (i)

If $\\{i,j\\}\cap\operatorname{\mathcal{S}}_{1}\neq\emptyset$, without loss of
generality, we can assume that $i\in\operatorname{\mathcal{S}}_{1}$ and
$j\in\operatorname{\mathcal{S}}_{2}$. Then, by (4.9),

$\displaystyle\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{i},X_{j}]$
$\displaystyle=\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|X_{i},X_{j}]\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{i}]$
$\displaystyle=\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|X_{i},X_{j}]\cdot
0=0.$

  2. (ii)

If $\\{i,j\\}\cap\operatorname{\mathcal{S}}_{1}=\emptyset$ and
$i\in\operatorname{\mathcal{S}}_{2}$ and $j\in\operatorname{\mathcal{S}}_{3}$,
then by (4.9),

$\displaystyle\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{i},X_{j}]$
$\displaystyle=\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|X_{i}]\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{j}]$
$\displaystyle=0\cdot 0=0.$

  3. (iii)

If $\\{i,j\\}\cap\operatorname{\mathcal{S}}_{1}=\emptyset$ and
$i,j\in\operatorname{\mathcal{S}}_{2}$, then

$\displaystyle\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})|X_{i},X_{j}]$
$\displaystyle=\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|X_{i},X_{j}]\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{3}})]$
$\displaystyle=\operatorname{\mathbb{E}}[\bar{h}_{m}(X_{\operatorname{\mathcal{S}}_{1}},X_{\operatorname{\mathcal{S}}_{2}})|X_{i},X_{j}]\cdot
0=0.$

∎

######  Lemma A.9 (Properties of $\delta_{2,b}$).

Assume $\operatorname{\mathbb{E}}[h(X_{1},\dots,X_{m})]=0$ and $\sigma^{2}=1$.
There exists a positive absolute constant $C(m)>0$ depending only on $m$ such
that

(A.25) $\|\delta_{2,b}\|_{2}\leq
C(m)\Bigg{\\{}\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{\\}}$

Moreover, for an absolute constant $C>0$,

(A.26) $|\operatorname{\mathbb{E}}[\bar{\delta}_{2,b}f_{x}(W_{b})]|\leq
Ce^{-x}\|\delta_{2,b}\|_{2}\text{ for }x\geq 1.$

###### Proof of Lemma A.9.

$\delta_{2,b}$ is precisely the quantity ”$\Pi_{22}$” in Leung and Shao (2023,
Appendix E.1), and the bound (A.25) is shown as equation $(E.3)$ in Leung and
Shao (2023) which states

$\|\delta_{2,b}\|_{2}^{2}\leq
C(m)\left(\frac{\operatorname{\mathbb{E}}[h^{2}]}{n}+\frac{\|g\|_{3}^{2}\|h\|_{3}^{2}}{n}\wedge\frac{\|h\|_{3}^{2}}{n^{2/3}}\right),$

and can be further simplified as (A.25) because $\|h\|_{2}\leq\|h\|_{3}$ and
$1=\|g\|_{2}\leq\|g\|_{3}$.

(A.26) can be easily proved using a technique from Leung and Shao (2023,
Appendix D.2) as follows: By (A.7) in Lemma A.2 and $0<f_{x}\leq 0.63$ in
Lemma A.1,

$\displaystyle\Big{|}\operatorname{\mathbb{E}}\Big{[}\bar{\delta}_{2,b}f_{x}(W_{b})\Big{]}\Big{|}$
$\displaystyle=\Big{|}\operatorname{\mathbb{E}}\Big{[}\bar{\delta}_{2,b}f_{x}(W_{b})I(W_{b}\leq
x-1)\Big{]}+\operatorname{\mathbb{E}}\Big{[}\bar{\delta}_{2,b}f_{x}(W_{b})I(W_{b}>x-1)\Big{]}\Big{|}$
$\displaystyle\leq
1.7e^{-x}\operatorname{\mathbb{E}}\Big{[}|\delta_{2,b}|\Big{]}+0.63\operatorname{\mathbb{E}}\Big{[}|\delta_{2,b}|\frac{e^{W_{b}}}{e^{x-1}}\Big{]}\leq
Ce^{-x}\|\delta_{2,b}\|_{2},$

where the last inequality uses Bennett’s inequality (Lemma A.5). ∎

##  Appendix B Proof of Lemma 4.1

###### Proof of Lemma 4.1$(i)$.

Rewrite $D_{1}$ in (4.12) as

$D_{1}=\frac{\sqrt{n}}{m}\times{n\choose m}^{-1}\sum_{1\leq
i_{1}<\dots<i_{m}\leq n}\bar{h}_{m}(X_{i_{1}},\dots,X_{i_{m}}).$

In light of (4.9), we recognize that ${n\choose m}^{-1}\sum_{1\leq
i_{1}<\dots<i_{m}\leq n}\bar{h}(X_{i_{1}},\dots,X_{i_{m}})$ is a mean-0
degenerate U-statistic of rank $2$. By the bound for the central absolute
moment of U-statistics in Lemma 4.4 and that
$\operatorname{\mathbb{E}}[|\bar{h}_{m}|^{3}]\leq
C(m)\operatorname{\mathbb{E}}[|h_{m}|^{3}]$ from (4.11), we have

$\operatorname{\mathbb{E}}[|D_{1}|^{3}]\leq\frac{n^{3/2}}{m^{3}}\frac{C(m)}{n^{3}}\operatorname{\mathbb{E}}[|h|^{3}]=\frac{C(m)}{n^{3/2}}\operatorname{\mathbb{E}}[|h|^{3}].$

By Markov’s inequality, we hence get

$P\Big{(}|D_{1}|>\frac{\mathfrak{c}_{m}x}{4}\Big{)}\leq\frac{64\operatorname{\mathbb{E}}[|D_{1}|^{3}]}{\mathfrak{c}_{m}^{3}x^{3}}\leq\frac{C(m)}{\mathfrak{c}_{m}^{3}n^{3/2}(1+x^{3})}\operatorname{\mathbb{E}}[|h|^{3}].$

∎

###### Proof of Lemma 4.1$(ii)$.

Let $W^{(i)}=W-\xi_{i}$, which satisfies

$P\Big{(}W^{(i)}\geq\frac{\mathfrak{c}_{m}x}{4}\Big{)}\leq P\left(\max_{1\leq
i\leq
n}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{6}\right)+e^{3/2}\Big{(}1+\frac{(\mathfrak{c}_{m}x)^{2}}{24}\Big{)}^{-3/2}$

by Chen et al. (2011, Lemma 8.2)(taking $p=3/2$ in that lemma). This implies

$\displaystyle P\Big{(}W\geq\frac{\mathfrak{c}_{m}x}{2},\max_{1\leq i\leq
n}|\xi_{i}|>1\Big{)}$
$\displaystyle\leq\sum_{i=1}^{n}P\Big{(}W\geq\frac{\mathfrak{c}_{m}x}{2},|\xi_{i}|>1\Big{)}$
$\displaystyle\leq\sum_{i=1}^{n}P\Big{(}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{4}\Big{)}+\sum_{i=1}^{n}P\Big{(}W^{(i)}\geq\frac{\mathfrak{c}_{m}x}{4}\Big{)}P(|\xi_{i}|>1)$
$\displaystyle\leq\sum_{i=1}^{n}P\Big{(}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{4}\Big{)}+\left\\{P\left(\max_{1\leq
i\leq
n}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{6}\right)+e^{3/2}\Big{(}1+\frac{(\mathfrak{c}_{m}x)^{2}}{24}\Big{)}^{-3/2}\right\\}\sum_{i=1}^{n}P(|\xi_{i}|>1)$
$\displaystyle\leq\sum_{i=1}^{n}P\Big{(}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{4}\Big{)}+\left\\{P\left(\max_{1\leq
i\leq
n}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{6}\right)+e^{3/2}\Big{(}1+\frac{(\mathfrak{c}_{m}x)^{2}}{24}\Big{)}^{-3/2}\right\\}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{2}I(|\xi_{i}|>1)]$
$\displaystyle\leq
2\sum_{i=1}^{n}P\Big{(}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{6}\Big{)}+e^{3/2}\Big{(}1+\frac{(\mathfrak{c}_{m}x)^{2}}{24}\Big{)}^{-3/2}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{2}I(|\xi_{i}|>1)]\text{
given \eqref{sigma_equal_1_assumption}}$ $\displaystyle\leq
2\sum_{i=1}^{n}P\Big{(}|\xi_{i}|>\frac{\mathfrak{c}_{m}x}{6}\Big{)}+\frac{(24\cdot
e)^{3/2}}{\mathfrak{c}_{m}^{3}(24+x^{2})^{3/2}}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{2}I(|\xi_{i}|>1)]\text{
given $0<\mathfrak{c}_{m}<1$}$
$\displaystyle\leq\frac{C}{\mathfrak{c}_{m}^{3}(1+x^{3})}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]=\frac{C\operatorname{\mathbb{E}}[|g|^{3}]}{\mathfrak{c}_{m}^{3}(1+x^{3})\sqrt{n}}.$

∎

###### Proof of Lemma 4.1$(iii)$.

$\displaystyle
P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{1}|>n^{-1/2}\right)$
$\displaystyle\leq
e^{-\mathfrak{c}_{m}x/2}\operatorname{\mathbb{E}}[e^{W_{b}}I(|\delta_{1}|>n^{-1/2})]$
$\displaystyle\leq
e^{-\mathfrak{c}_{m}x/2}\|e^{W_{b}}\|_{3}\|I(|\delta_{1}|>n^{-1/2})\|_{3/2}\text{
by H\"{o}lder's inequality}$ $\displaystyle\leq
Ce^{-\mathfrak{c}_{m}x/2}\|n^{1/2}|\delta_{1}|\|_{3/2}\text{ by Bennett's
inequality (Lemma~{}\ref{lem:bennett_censored}) and
$I(|\delta_{1}|>n^{-1/2})\leq n^{1/2}|\delta_{1}|$}$ $\displaystyle\leq
C(m)e^{-\mathfrak{c}_{m}x/2}\frac{\|h(X_{1},\dots,X_{m})\|_{3}^{2}}{\sqrt{n}}\text{
by \eqref{delta1_3over2_th_abs_moment_bdd} in
Lemma~{}\ref{lem:delta1_moment_bdd} }.$

∎

###### Proof of Lemma 4.1$(iv)$.

$\displaystyle
P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|\delta_{2,b}|>1\right)$
$\displaystyle\leq\operatorname{\mathbb{E}}[e^{W_{b}-\frac{\mathfrak{c}_{m}x}{2}}I(|\delta_{2,b}|>1)]$
$\displaystyle\leq
e^{-\mathfrak{c}_{m}x/2}\operatorname{\mathbb{E}}[e^{W_{b}}|\delta_{2,b}|]$
$\displaystyle\leq Ce^{-\mathfrak{c}_{m}x/2}\|\delta_{2,b}\|_{2}\text{ by
Bennett's inequality (Lemma~{}\ref{lem:bennett_censored}) }$
$\displaystyle\leq
C(m)e^{-\mathfrak{c}_{m}x/2}\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\text{ by
Lemma~{}\ref{lem:delta2b_property}}.$

∎

###### Proof of Lemma 4.1$(v)$.

From the alternative form of $D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}$
in (A.10),

$\displaystyle
P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right)$
$\displaystyle\leq
e^{-\mathfrak{c}_{m}x/2}\operatorname{\mathbb{E}}[e^{W_{b}}D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}^{2}]$
$\displaystyle\leq
Ce^{-\mathfrak{c}_{m}x/2}\Bigg{\\{}\operatorname{\mathbb{E}}[e^{W_{b}}\delta_{0,b}^{2}]+\operatorname{\mathbb{E}}[e^{W_{b}}\bar{\delta}_{1}^{2}]+\operatorname{\mathbb{E}}[e^{W_{b}}\bar{\delta}_{2,b}^{2}]$
$\displaystyle\hskip
28.45274pt+\operatorname{\mathbb{E}}[e^{W_{b}}]\Bigg{(}\frac{\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]}{n-1}+\sum_{i=1}^{n}\operatorname{\mathbb{E}}\Big{[}(\xi_{i}^{2}-1)I(|\xi_{i}|>1)\Big{]}\Bigg{)}\Bigg{\\}},$

where we have also used that $d_{n}\leq 2$ and both

$\frac{\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]}{n-1}\text{ and
}\sum_{i=1}^{n}\operatorname{\mathbb{E}}\Big{[}(\xi_{i}^{2}-1)I(|\xi_{i}|>1)\Big{]}$

are less than 1. Continuing, we get

$\displaystyle
P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right)$
$\displaystyle\leq
Ce^{-\mathfrak{c}_{m}x/2}\\{\operatorname{\mathbb{E}}[e^{W_{b}}\delta_{0,b}^{2}]+\|e^{W_{b}}\|_{2}\|\bar{\delta}_{1}^{2}\|_{2}+\|e^{W_{b}}\|_{2}\|\bar{\delta}_{2,b}^{2}\|_{2}+n^{-1}+\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{2}I(|\xi_{i}|>1)]\\}$
(by Lemma A.5) $\displaystyle\leq
Ce^{-\mathfrak{c}_{m}x/2}\Big{\\{}\operatorname{\mathbb{E}}[e^{W_{b}}\delta_{0,b}^{2}]+\|e^{W_{b}}\|_{2}\sqrt{\operatorname{\mathbb{E}}[|\bar{\delta}_{1}|]}+\|e^{W_{b}}\|_{2}\|\bar{\delta}_{2,b}\|_{2}+n^{-1}+\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]\Big{\\}}$
$\displaystyle\hskip 284.52756pt\text{ (by
$|\bar{\delta}_{1}|\vee|\bar{\delta}_{2,b}|\leq 1$)}.$

To wrap up the proof, apply Bennett’s inequality (Lemma A.5), Lemmas A.7\-
A.9, $n^{-1}\leq n^{-1/2}\operatorname{\mathbb{E}}[|g|^{3}]$ and that
$\|h\|_{2}\leq\|g\|_{3}\|h\|_{3}$ to the last line and get

$P\left(W_{b}\geq\frac{\mathfrak{c}_{m}x}{2},|D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}|>1\right)\leq
Ce^{-\mathfrak{c}_{m}x/2}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]+\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{)}.$

∎

###### Proof of (4.27).

Given (4.26), simply putting $(i)-(v)$ together and use the fact
$\|h\|_{3}\|g\|_{3}\leq\|h\|_{3}^{2}$. ∎

##  Appendix C Proof of Lemma 4.2

Before proving Lemma 4.2, we shall review a useful property of variable
censoring discussed in Leung and Shao (2023, Section 2.0.3):

######  Property C.1.

Suppose $Y$ and $Z$ are any two real-value variables censored in the _same
manner_ , i.e. for some
$a,b\in\operatorname{\mathbb{R}}\cup\\{-\infty,\infty\\}$ with $a\leq b$, we
define their censored versions

$\bar{Y}\equiv aI(Y<a)+YI(a\leq Y\leq b)+bI(Y>b)$

and

$\bar{Z}\equiv aI(Z<a)+ZI(a\leq Z\leq b)+bI(Z>b).$

Then it must be that $|\bar{Y}-\bar{Z}|\leq|Y-Z|.$

Now we begin the proof. We first let

$D_{2,b,n^{-1/2},\bar{\delta}_{2,b}}=d_{n}^{2}(V_{n,b}^{2}+n^{-1/2}+\bar{\delta}_{2,b})-1\text{
and
}D_{2,b,-n^{-1/2},\bar{\delta}_{2,b}}=d_{n}^{2}(V_{n,b}^{2}-n^{-1/2}+\bar{\delta}_{2,b})-1,$

where we respectively replaced $\bar{\delta}_{1}$ with its lower and upper
bounds $-n^{-1/2}$ and $n^{-1/2}$ from the definition of
$D_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}$ in (4.22). Analogously to
(4.23), we also let

$\bar{D}_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}\equiv
D_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq
D_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}\leq 1\Bigg{)}\\\
+I\Bigg{(}D_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}D_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}.$

and

$\bar{D}_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}\equiv
D_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq
D_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}\leq 1\Bigg{)}\\\
+I\Bigg{(}D_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}D_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}.$

With respect to these, we define the ”placeholder” denominator remainder

(C.1) $\mathfrak{D}_{2}=\mathfrak{D}_{2}(X_{1},\dots,X_{n})\equiv
d_{n}^{2}(V_{n,b}^{2}+(-n^{-1/2}|n^{-1/2})+\bar{\delta}_{2,b})-1\\\
=d_{n}^{2}\Bigg{(}\delta_{0,b}+(-n^{-\frac{1}{2}}|n^{-\frac{1}{2}})+\bar{\delta}_{2,b}\Bigg{)}+\frac{\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{b,i}^{2}]}{n-1}-\sum_{i=1}^{n}\operatorname{\mathbb{E}}\Big{[}(\xi_{i}^{2}-1)I(|\xi_{i}|>1)\Big{]}$

(where the second line comes from (A.10)) and its censored version

(C.2)
$\bar{\mathfrak{D}}_{2}\equiv\mathfrak{D}_{2}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq\mathfrak{D}_{2}\leq
1\Bigg{)}+I\Bigg{(}\mathfrak{D}_{2}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}\mathfrak{D}_{2}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)},$

where for any $a,b\in\operatorname{\mathbb{R}}$, $(a|b)$ represents either $a$
or $b$, which means that

$\mathfrak{D}_{2}\text{ represents either
}D_{2,b,n^{-1/2},\bar{\delta}_{2,b}}\text{ or
}D_{2,b,-n^{-1/2},\bar{\delta}_{2,b}}.$

Since $-n^{-1/2}\leq\bar{\delta}_{1}\leq n^{-1/2}$ by its definition (4.21),
it is easy to see that

(C.3)
$P(W_{b}+\bar{D}_{1,x}>x(1+\bar{D}_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}})^{1/2})\\\
\leq
P(W_{b}+\bar{D}_{1,x}>x(1+\bar{D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}})^{1/2})\\\
\leq
P(W_{b}+\bar{D}_{1,x}>x(1+\bar{D}_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}})^{1/2}).$

Therefore, to show Lemma 4.2, it suffices to show the same bound (4.28) with
$\bar{D}_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}$ replaced by
$\bar{D}_{2,V_{b},n^{-1/2},\bar{\delta}_{2,b}}$ or
$\bar{D}_{2,V_{b},-n^{-1/2},\bar{\delta}_{2,b}}$, i.e.,

(C.4)
$\Big{|}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\Big{)}-\bar{\Phi}(x)\Big{|}\leq\frac{C(m)}{e^{c(m)x}}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}^{2}\|h\|_{3}}{\sqrt{n}}\Bigg{)}.$

As will be seen later, transforming the problem into one that handles
$\mathfrak{D}_{2}$ instead of
${D}_{2,V_{b},\bar{\delta}_{1},\bar{\delta}_{2,b}}$ has the advantage of
obviating the need to deal with the variability of $\bar{\delta}_{1}$; a
similar strategy has also been employed in our related work Leung and Shao
(2023) for proving uniform B-E bounds.

Since $\bar{\mathfrak{D}}_{2}>-1$ almost surely, by applying the elementary
inequality that

$(1+s)^{1/2}\leq 1+s/2\text{ for all }s\geq-1,$

one get the two event inclusions

$\Bigg{\\{}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\Bigg{\\}}\\\
\subset\left\\{W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}>x\right\\}\bigcup\left\\{x(1+\bar{\mathfrak{D}}_{2})^{1/2}<W_{b}+\bar{D}_{1,x}\leq
x(1+\bar{\mathfrak{D}}_{2}/2)\right\\}$

and

$\Bigg{\\{}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\Bigg{\\}}\supset\left\\{W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}>x\right\\}.$

These imply

(C.5)
$\Big{|}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\Big{)}-\bar{\Phi}(x)\Big{|}\leq\\\
P\left(x(1+\bar{\mathfrak{D}}_{2})^{1/2}<W_{b}+\bar{D}_{1,x}\leq
x(1+\bar{\mathfrak{D}}_{2}/2)\right)+\left|P\left(W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}>x\right)-\bar{\Phi}(x)\right|.$

Hence, proving (C.4) boils down to bounding the two terms on the right of
(C.5)

To do this, we shall first define the ”leave-one-out” variants for some of the
variables involved. Let $i\in\\{1,\dots,n\\}$ be any sample point. For the
numerator remainder, we define the variant of $D_{1}$ with all terms involving
$X_{i}$ omitted, i.e.

(C.6)
$D_{1}^{(i)}=D_{1}^{(i)}(X_{1},\dots,X_{i-1},X_{i}\dots,X_{n})\equiv{n-1\choose
m-1}^{-1}\sum_{\begin{subarray}{c}1\leq i_{1}<\dots<i_{m}\leq n\\\ i_{l}\neq
i\text{ for
}l=1,\dots,m\end{subarray}}\frac{\bar{h}_{m}(X_{i_{1}},X_{i_{2}},\dots,X_{i_{m}})}{\sqrt{n}},$

and its corresponding censored version

$\bar{D}_{1,x}^{(i)}\equiv
D_{1}^{(i)}I\left(|D_{1}^{(i)}|\leq\frac{\mathfrak{c}_{m}x}{4}\right)+\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1}^{(i)}>\frac{\mathfrak{c}_{m}x}{4}\right)-\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1}^{(i)}<-\frac{\mathfrak{c}_{m}x}{4}\right).$

For the denominator remainder, we first define

$\delta_{2,b}^{(i)}\equiv\frac{2(n-1)}{\sqrt{n}(n-m)}{n-1\choose
m-1}^{-1}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\xi_{b,j}\sum_{\begin{subarray}{c}1\leq
i_{1}<\dots<i_{m-1}\leq n\\\ i_{l}\neq j,i\text{ for
}l=1,\dots,m-1\end{subarray}}\bar{h}_{m}(X_{j},X_{i_{1}},\dots,X_{i_{m-1}}).$

and its censored variant

$\bar{\delta}_{2,b}^{(i)}=\delta_{2,b}^{(i)}I(|\delta_{2,b}^{(i)}|\leq
1)+I(\delta_{2,b}^{(i)}>1)-I(\delta_{2,b}^{(i)}<-1).$

Base on them, we can define the ”leave-one-out” denominator remainder

(C.7)
$\mathfrak{D}_{2}^{(i)}=\mathfrak{D}_{2}^{(i)}(X_{1},\dots,X_{i-1},X_{i},\dots,X_{n})\equiv
d_{n}^{2}\Big{(}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\xi_{b,j}^{2}+(-n^{-1/2}|n^{-1/2})+\bar{\delta}_{2,b}^{(i)}\Big{)}-1$

that omits all terms involving $X_{i}$ or $\xi_{i}$, and its censored version

$\bar{\mathfrak{D}}_{2}^{(i)}\equiv\mathfrak{D}_{2}^{(i)}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq\mathfrak{D}_{2}^{(i)}\leq
1\Bigg{)}+I\Bigg{(}\mathfrak{D}_{2}^{(i)}>1\Bigg{)}+\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}I\Bigg{(}\mathfrak{D}_{2}^{(i)}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}.$

With these notions, we can state the bounds for the right-hand-side terms of
(C.5).

######  Lemma C.2 (Randomized concentration inequality).

Let $W$, $D_{1}$, $D_{2}$ be as defined in Section 4 for $T_{n}$ and
$T_{n}^{*}$. Under the assumptions of Theorem 3.1 and (4.2), for any $x\geq
1$,

$P\left(x\left(1+\bar{\mathfrak{D}}_{2}\right)^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq x\left(1+\bar{\mathfrak{D}}_{2}/2\right)\right)\leq
Cxe^{-\mathfrak{c}_{m}x/4}\times\\\
\Bigg{\\{}\operatorname{\mathbb{E}}\Big{[}(1+e^{W_{b}})\bar{\mathfrak{D}}_{2}^{2}\Big{]}+\sum_{i=1}^{n}\Big{(}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\Big{\|}\xi_{i}\Big{\|}_{2}\Big{\|}\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\Big{\|}_{2}+\Big{\|}\xi_{i}\Big{\|}_{3}\Big{\|}\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\Big{\|}_{3/2}\Big{)}\Bigg{\\}}$

where $D_{1}^{(i)}$, $D_{2}^{(i)}$ are random variables such that $\xi_{i}$ is
independent of $(W-\xi_{i},D_{1}^{(i)},D_{2}^{(i)})$.

######  Lemma C.3 (Nonuniform Berry-Esseen bound for
$W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}$).

Assuming $\max_{1\leq i\leq n}\|\xi_{i}\|_{3}<\infty$, for any $x\geq 1$,

(C.8)
$\left|P\left(W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}>x\right)-\bar{\Phi}(x)\right|\leq
x\Bigg{|}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]\Bigg{|}+C(m)e^{-c(m)x}\times\\\
\Bigg{\\{}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\sum_{i=1}^{n}\Big{(}\|\xi_{i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\|\xi_{i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}\Big{)}+\|\bar{D}_{1,x}\|_{2}+\operatorname{\mathbb{E}}[(1+e^{W_{b}})\mathfrak{D}_{2}^{2}]\Bigg{\\}}$

Combining Lemmas C.2 and C.3 with (C.5), we get

(C.9)
$\Big{|}P\Big{(}W_{b}+\bar{D}_{1,x}>x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\Big{)}-\bar{\Phi}(x)\Big{|}\leq
x\Bigg{|}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]\Bigg{|}+C(m)e^{-c(m)x}\times\\\
\Bigg{\\{}\operatorname{\mathbb{E}}\Big{[}(1+e^{W_{b}})\mathfrak{D}_{2}^{2}\Big{]}+\|D_{1}\|_{2}+\sum_{i=1}^{n}\Bigg{(}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\Big{\|}\xi_{i}\Big{\|}_{2}\Big{\|}D_{1}-D_{1}^{(i)}\Big{\|}_{2}+\Big{\|}\xi_{i}\Big{\|}_{3}\Big{\|}\mathfrak{D}_{2}-\mathfrak{D}_{2}^{(i)}\Big{\|}_{3/2}\Bigg{)}\Bigg{\\}}$

by Property C.1 and $\|\bar{D}_{1,x}\|\leq\|D_{1}\|_{2}$. At this point, we
define the typical quantities:

$\quad\beta_{2}\equiv\sum_{i=1}^{n}\mathbb{E}[\xi_{i}^{2}I(|\xi_{i}|>1)]\quad\text{
and
}\quad\beta_{3}\equiv\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{3}I(|\xi_{i}|\leq
1)],$

which has the property
$\beta_{2}+\beta_{3}\leq\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]$.
The following bounds allow us to arrive at (C.4) from (C.9),

(C.10)
$\|D_{1}\|_{2}\leq\frac{(m-1)\|h\|_{2}}{\sqrt{m(n-m+1)}}\leq\frac{C(m)\|h\|_{2}}{\sqrt{n}}.$
(C.11)
$\|D_{1}-D_{1}^{(i)}\|_{2}\leq\frac{\sqrt{2}(m-1)\|h\|_{2}}{\sqrt{nm(n-m+1)}}\leq\frac{C(m)\|h\|_{2}}{n}.$
(C.12) $\operatorname{\mathbb{E}}[\mathfrak{D}_{2}^{2}]\leq
C(m)\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{)}$
(C.13) $\operatorname{\mathbb{E}}[e^{W_{b}}\mathfrak{D}_{2}^{2}]\leq
C(m)\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{)}$
(C.14)
$\bigg{|}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]\bigg{|}\leq
C(m)e^{-x}\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{)}$
(C.15) $\|\mathfrak{D}_{2}-\mathfrak{D}_{2}^{(i)}\|_{3/2}\leq
C(m)\Bigg{(}\frac{\|g\|_{3}^{2}}{n}+\frac{\|g\|_{3}\|h\|_{3}}{n}\Bigg{)}$

These bounds are proved as follows:

(C.10) and (C.11): The proofs can be found in Chen et al. (2011, Lemma 10.1).

(C.12): From (C.1), we have

$\displaystyle\operatorname{\mathbb{E}}[\mathfrak{D}_{2}^{2}]$
$\displaystyle\leq
C\operatorname{\mathbb{E}}[\delta_{0,b}^{2}+\bar{\delta}_{2,b}^{2}+n^{-1}+\beta_{2}^{2}]$
$\displaystyle\leq
C\operatorname{\mathbb{E}}[\delta_{0,b}^{2}+|\delta_{2,b}|+n^{-1}+\beta_{2}]\text{
since $\bar{\delta}_{2,b},\beta_{2}\leq 1$}$ $\displaystyle\leq
C(\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\|\delta_{2,b}\|_{2}+n^{-1})\text{
by \eqref{bdd_exp_delta0b_sq},
$\beta_{2}\leq\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]$ }$
$\displaystyle\leq
C(m)\Bigg{(}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{)}\text{
by \eqref{delta2b_l2_bdd} and $\|g\|_{3}\|h\|_{3}\geq 1$}.$

(C.13): From (C.1) again, we have

$\displaystyle\operatorname{\mathbb{E}}[e^{W_{b}}\mathfrak{D}_{2}^{2}]$
$\displaystyle\leq
C\operatorname{\mathbb{E}}[e^{W_{b}}(\delta_{0,b}^{2}+|\delta_{2,b}|+n^{-1}+\beta_{2})]\text{
since $\bar{\delta}_{2,b},\beta_{2}\leq 1$}$ $\displaystyle\leq
C\Big{(}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\|\delta_{2,b}\|_{2}+n^{-1}\Big{)}$
by (A.12), Bennett’s inequality (Lemma A.5) and
$\beta_{2}\leq\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]$
$\displaystyle\leq
C(m)\Big{\\{}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Big{\\}}\text{
by \eqref{delta2b_l2_bdd}}$

(C.14): From (C.1) again, we get

$\displaystyle\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]$
$\displaystyle\leq
C\Big{\\{}|\operatorname{\mathbb{E}}[f_{x}(W_{b})(\delta_{0,b}+\bar{\delta}_{2,b})]|+e^{-x}(n^{-1/2}+\beta_{2})\Big{\\}}\text{
by Lemma~{}\ref{lem:expect_fxWB}}$ $\displaystyle\leq
C(m)e^{-x}\Bigg{\\{}\frac{\operatorname{\mathbb{E}}[|g|^{3}]}{\sqrt{n}}+\frac{\|g\|_{3}\|h\|_{3}}{\sqrt{n}}\Bigg{\\}}\text{
by \eqref{bdd_fx_delta0b}, \eqref{delta2b_l2_bdd} and \eqref{bdd_fx_delta2b}}$

(C.15): Since
$\mathfrak{D}_{2}-\mathfrak{D}_{2}^{(i)}=d_{n}^{2}(\xi_{b,i}^{2}+\bar{\delta}_{2,b}-\bar{\delta}_{2,b}^{(i)})$,

$\|\mathfrak{D}_{2}-\mathfrak{D}_{2}^{(i)}\|_{3/2}\leq
C\Bigg{\\{}(\operatorname{\mathbb{E}}[|\xi_{b,i}|^{3}])^{2/3}+\|\delta_{2,b}-\delta_{2,b}^{(i)}\|_{3/2}\Bigg{\\}}.$

In our related work Leung and Shao (2023, Appendix E), we have already shown
that

$\|\delta_{2,b}-\delta_{2,b}^{(i)}\|_{3/2}\leq\frac{C(m)\|g\|_{3}\|h\|_{3}}{n},$

so we get

$\|\mathfrak{D}_{2}-\mathfrak{D}_{2}^{(i)}\|_{3/2}\leq
C(m)\Bigg{(}\frac{\|g\|_{3}^{2}}{n}+\frac{\|g\|_{3}\|h\|_{3}}{n}\Bigg{)}.$

(Note that $\delta_{2,b}-\delta_{2,b}^{(i)}$ is precisely the quantity ”$A+B$”
appearing in Leung and Shao (2023, Appendix E.2))

It remains to prove Lemmas C.2 and C.3, which is the focus next.

###  C.1. Proof of Lemma C.2

If $\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]\geq 2$, we will
have

$\displaystyle P\left(x\left(1+\bar{\mathfrak{D}}_{2}\right)^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq x\left(1+\bar{\mathfrak{D}}_{2}/2\right)\right)$
$\displaystyle\leq P\Big{(}\frac{3\mathfrak{c}_{m}x}{4}\leq
W_{b}+\bar{D}_{1,x}\Big{)}$ $\displaystyle\leq
P\Big{(}W_{b}\geq\frac{\mathfrak{c}_{m}x}{2}\Big{)}\leq
e^{-\mathfrak{c}_{m}x/2}\mathbb{E}[e^{W_{b}}]\leq
Ce^{-\mathfrak{c}_{m}x/2}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}],$

since the Bennett’s inequality (Lemma A.5) implies $\mathbb{E}[e^{W_{b}}]\leq
C\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]$ for some absolute constant $C>0$,
and Lemma C.2 follows because $x\geq 1$.

If $\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]<2$, since
$x(1+\bar{\mathfrak{D}}_{2})^{1/2}\geq\frac{3\mathfrak{c}_{m}x}{4}$, it must
be less that

$\displaystyle P\left(x\left(1+\bar{\mathfrak{D}}_{2}\right)^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq x\left(1+\bar{\mathfrak{D}}_{2}/2\right)\right)$
$\displaystyle
e^{-(3\mathfrak{c}_{m}x)/8}\mathbb{E}\left[e^{(W_{b}+\bar{D}_{1,x})/2}I\left\\{x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq
x\Big{(}1+\bar{\mathfrak{D}}_{2}/2\Big{)}\right\\}\right]$ $\displaystyle\leq
e^{-(\mathfrak{c}_{m}x)/4}\mathbb{E}\left[e^{W_{b}/2}I\left\\{x\Big{(}1+\bar{\mathfrak{D}}_{2}\Big{)}^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq
x\Big{(}1+\bar{\mathfrak{D}}_{2}/2\Big{)}\right\\}\right]$ (C.16)
$\displaystyle\leq
e^{-(\mathfrak{c}_{m}x)/4}\operatorname{\mathbb{E}}\Bigg{[}e^{W_{b}/2}I\Big{(}x+\frac{x(\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{2})}{2}-\bar{D}_{1,x}\leq
W_{b}\leq x+\frac{x\bar{\mathfrak{D}}_{2}}{2}-\bar{D}_{1,x}\Big{)}\Bigg{]},$

the last inequality follows from the fact that

$1+s/2-s^{2}/2\leq(1+s)^{1/2}\text{ for all }s\geq-1.$

Continuing from (C.16), by the exponential randomized concentration inequality
for a sum of censored variables (Lemma A.6), we have

$\displaystyle
e^{(\mathfrak{c}_{m}x)/4}P\left(x\left(1+\bar{\mathfrak{D}}_{2}\right)^{1/2}\leq
W_{b}+\bar{D}_{1,x}\leq x\left(1+\bar{\mathfrak{D}}_{2}/2\right)\right)$
$\displaystyle\leq\left(\mathbb{E}\left[e^{W_{b}}\right]\right)^{1/2}\exp\left(-\frac{1}{16(\beta_{2}+\beta_{3})^{2}}\right)$
$\displaystyle+Ce^{(\beta_{2}+\beta_{3})/8}\Biggl{\\{}\sum_{i=1}^{n}\mathbb{E}[|\xi_{b,i}|e^{W_{b}^{(i)}/2}(|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}|+x|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}|)]$
$\displaystyle+\mathbb{E}\left[|W_{b}|e^{W_{b}/2}\left(x\bar{\mathfrak{D}}_{2}^{2}+\beta_{2}+\beta_{3}\right)\right]$
(C.17)
$\displaystyle+\sum_{i=1}^{n}\Big{|}\operatorname{\mathbb{E}}[\xi_{b,i}]\Big{|}\operatorname{\mathbb{E}}\left[e^{W_{b}^{(i)}/2}\left(x(\bar{\mathfrak{D}}_{2}^{(i)})^{2}+\beta_{2}+\beta_{3}\right)\right]\Biggr{\\}},$

where we have used the fact that
$|\bar{\mathfrak{D}}_{2}|\vee|\bar{\mathfrak{D}}_{2}^{(i)}|\leq 1$, which
implies

$\displaystyle|\bar{\mathfrak{D}}_{2}^{2}-(\bar{\mathfrak{D}}_{2}^{(i)})^{2}|$
$\displaystyle=|(\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)})(\bar{\mathfrak{D}}_{2}+\bar{\mathfrak{D}}_{2}^{(i)})|\leq
2|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}|.$

It remains to bound the terms on the right hand side of (C.17).

First,

(C.18) $\operatorname{\mathbb{E}}[e^{W_{b}}]\leq C\text{ for some }C>0\text{
by Bennett's inequality (Lemma~{}\ref{lem:bennett_censored})}$

and

(C.19) $\displaystyle\exp\left(\frac{-1}{16(\beta_{2}+\beta_{3})^{2}}\right)$
$\displaystyle\leq C(\beta_{2}+\beta_{3})\leq
C\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}].$

Secondly,

$\displaystyle\mathbb{E}[|\xi_{b,i}|e^{W_{b}^{(i)}/2}(|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}|+x|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}|)]$
$\displaystyle\leq\|\xi_{b,i}e^{W_{b}^{(i)}/2}\|_{2}\|\bar{D}_{1,x}-{D}_{1,x}^{(i)}\|_{2}+x\|\xi_{b,i}e^{W_{b}^{(i)}/2}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)})\|_{3/2}$
(C.20) $\displaystyle\leq
C\Big{\\{}\Big{\|}\xi_{b,i}\Big{\|}_{2}\Big{\|}{D}_{1,x}-{D}_{1,x}^{(i)}\Big{\|}_{2}+x\|\xi_{b,i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)})\|_{3/2}\Big{\\}},$

where the last inequality uses that for any $2\leq p\leq 3$,
$\|\xi_{b,i}e^{W_{b}^{(i)}/2}\|_{p}\leq C\|\xi_{b,i}\|_{p}$ by the
independence of $\xi_{b,i}$ and $e^{W_{b}^{(i)}/2}$, as well as Bennett’s
inequality (Lemma A.5). Thirdly, since $|W_{b}|/2\leq e^{|W_{b}|/2}\leq
e^{W_{b}/2}+e^{-W_{b}/2}$, by the Bennett’s inequality (Lemma A.5) again,

(C.21)
$\mathbb{E}\left[|W_{b}|e^{W_{b}/2}\left(x\bar{\mathfrak{D}}_{2}^{2}+\beta_{2}+\beta_{3}\right)\right]\leq
C\Big{(}x\operatorname{\mathbb{E}}[(1+e^{W_{b}})\bar{\mathfrak{D}}_{2}^{2}]+\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]\Big{)}$

Lastly, by Lemma A.4, Bennett’s inequality (Lemma A.5) and the current
assumption that
$\beta_{2}+\beta_{3}\leq\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]<2$,
we have

(C.22)
$\sum_{i=1}^{n}\Bigg{|}\operatorname{\mathbb{E}}[\xi_{b,i}]\Bigg{|}\operatorname{\mathbb{E}}\left[e^{W_{b}^{(i)}/2}\underbrace{\left(x(\bar{\mathfrak{D}}_{2}^{(i)})^{2}+\beta_{2}+\beta_{3}\right)}_{\leq
Cx\text{ as }x\geq 1}\right]\leq
Cx\sum_{i=1}^{n}\operatorname{\mathbb{E}}[\xi_{i}^{3}].$

Combining (C.17)-(C.22) with $x\geq 1$, Lemma C.2 is proved when
$\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]<2$.

###  C.2. Proof of Lemma C.3

We shall equivalently bound

(C.23) $|P(W_{b}+\bar{D}_{1,x}-\frac{x}{2}\bar{\mathfrak{D}}_{2}\leq
x)-\Phi(x)|.$

We first let $X_{1}^{*},\dots,X_{n}^{*}$ be independent copies of
$X_{1},\dots,X_{n}$ and define

$D_{1,i^{*}}\equiv
D_{1}(X_{1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{n})\text{ and }\\\
\mathfrak{D}_{2,i^{*}}\equiv\mathfrak{D}_{2}(X_{1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{n})\text{
for each }i\in\\{1,\dots,n\\},$

which are versions of $D_{1}$ and $\mathfrak{D}_{2}$ with $X_{i}^{*}$
replacing $X_{i}$ as input. In analogy to (4.19) and (C.2), also define their
correspondingly censored versions

$\bar{D}_{1,i^{*},x}\equiv
D_{1,i^{*}}I\left(|D_{1,i^{*}}|\leq\frac{\mathfrak{c}_{m}x}{4}\right)+\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1,i^{*}}>\frac{\mathfrak{c}_{m}x}{4}\right)-\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1,i^{*}}<-\frac{\mathfrak{c}_{m}x}{4}\right)\\\
\text{ and }\\\
\bar{\mathfrak{D}}_{2,i^{*}}\equiv\mathfrak{D}_{2,i^{*}}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq\mathfrak{D}_{2,i^{*}}\leq
1\Bigg{)}+I\Bigg{(}\mathfrak{D}_{2,i^{*}}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}\mathfrak{D}_{2,i^{*}}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}$

By letting

(C.24) $\Delta\equiv\bar{D}_{1,x}-\frac{x\bar{\mathfrak{D}}_{2}}{2}\text{ and
}\Delta_{i^{*}}\equiv\bar{D}_{1,i^{*},x}-\frac{x\bar{\mathfrak{D}}_{2,i^{*}}}{2},$

one can write the difference in (C.23) as

$\displaystyle P(W_{b}+\Delta\leq x)-\Phi(x)=\;$
$\displaystyle\mathbb{E}\left[f_{x}^{\prime}(W_{b}+\Delta)\right]-\mathbb{E}\left[W_{b}f_{x}(W_{b}+\Delta)\right]-\mathbb{E}\left[\Delta
f_{x}(W_{b}+\Delta)\right]$ (C.25) $\displaystyle=\;$ $\displaystyle
E_{1}+E_{2}+E_{3}$

where

$\displaystyle E_{1}$
$\displaystyle\equiv\sum_{i=1}^{n}\mathbb{E}\left[\int_{-1}^{1}\left\\{f_{x}^{\prime}(W_{b}+\Delta)-f_{x}^{\prime}(W_{b}^{(i)}+\Delta_{i^{*}}+t)\right\\}k_{b,i}(t)dt\right]$
$\displaystyle E_{2}$
$\displaystyle\equiv\sum_{i=1}^{n}\mathbb{E}\left[\xi_{b,i}\left\\{f_{x}(W_{b}+\Delta_{i^{*}})-f_{x}(W_{b}+\Delta)\right\\}\right],$
$\displaystyle E_{3}$
$\displaystyle\equiv-\sum_{i=1}^{n}\mathbb{E}[\xi_{b,i}f_{x}(W_{b}^{(i)}+\Delta_{i^{*}})]+\mathbb{E}[f_{x}^{\prime}(W_{b}+\Delta)]\sum_{i=1}^{n}\operatorname{\mathbb{E}}[(\xi_{i}^{2}-1)I(|\xi_{i}|>1)]-\mathbb{E}[\Delta
f_{x}(W_{b}+\Delta)],$

with $k_{b,i}$ defined as

$k_{b,i}(t)\equiv\mathbb{E}\left[\xi_{b,i}\\{I(0\leq
t\leq\xi_{b,i})-I(\xi_{b,i}\leq t<0)\\}\right],$

which has the properties

(C.26)
$\int_{-1}^{1}k_{b,i}i(t)dt=\mathbb{E}[\xi_{b,i}^{2}]\leq\operatorname{\mathbb{E}}[\xi_{i}^{2}]\quad\text{
and
}\quad\int_{-1}^{1}|t|k_{b,i}(t)dt=\frac{\mathbb{E}[|\xi_{b,i}|^{3}]}{2}\leq\frac{\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]}{2}.$

We will establish that

(C.27) $|E_{1}|\leq
Ce^{-c(m)x}\Bigg{\\{}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\sum_{i=1}^{n}\Big{(}\|\xi_{i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\|\xi_{i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}\Big{)}\Bigg{\\}},$
(C.28) $|E_{2}|\leq
Ce^{-cx}\sum_{i=1}^{n}\Bigg{(}\|\xi_{b,i}\|_{2}\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\|\xi_{b,i}\|_{3}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}\Bigg{)}.$

and

(C.29) $|E_{3}|\leq
C(m)e^{-cx}\Bigg{(}\sum_{i=1}^{n}\operatorname{\mathbb{E}}[|\xi_{i}|^{3}]+\|\bar{D}_{1,x}\|_{2}+\operatorname{\mathbb{E}}[(1+e^{W_{b}})\mathfrak{D}_{2}^{2}]\Bigg{)}+x\Big{|}\operatorname{\mathbb{E}}[\mathfrak{D}_{2}f_{x}(W_{b})]\Big{|},$

from which Lemma C.3 can be concluded.

Define, for any pair $1\leq i,j\leq n$,

$D_{1,i^{*}}^{(j)}\equiv\begin{cases}D_{1}^{(j)}(X_{1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{j-1},X_{j+1},\dots,X_{n})&\text{if
}i<j;\\\
D_{1}^{(j)}(X_{1},\dots,X_{j-1},X_{j+1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{n})&\text{if
}j<i;\\\ D_{1}^{(j)}(X_{1},\dots,X_{i-1},X_{i+1},\dots,X_{n})&\text{if
}i=j,\\\ \end{cases}$

and

$\mathfrak{D}_{2,i^{*}}^{(j)}\equiv\begin{cases}\mathfrak{D}_{2}^{(j)}(X_{1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{j-1},X_{j+1},\dots,X_{n})&\text{if
}i<j;\\\
\mathfrak{D}_{2}^{(j)}(X_{1},\dots,X_{j-1},X_{j+1},\dots,X_{i-1},X_{i}^{*},X_{i+1},\dots,X_{n})&\text{if
}j<i;\\\
\mathfrak{D}_{2}^{(j)}(X_{1},\dots,X_{i-1},X_{i+1},\dots,X_{n})&\text{if
}i=j,\\\ \end{cases}$

i.e., $D_{1,i^{*}}^{(j)}$ and $\mathfrak{D}_{2,i^{*}}^{(j)}$ are versions of
$D_{1}^{(j)}$ and $\mathfrak{D}_{2}^{(j)}$ with $X_{i}^{*}$ replacing $X_{i}$
as input; likewise, they have their censored variants

$\bar{D}_{1,i^{*},x}^{(j)}\equiv
D_{1,i^{*}}^{(j)}I\left(|D_{1,i^{*}}^{(j)}|\leq\frac{\mathfrak{c}_{m}x}{4}\right)+\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1,i^{*}}^{(j)}>\frac{\mathfrak{c}_{m}x}{4}\right)-\frac{\mathfrak{c}_{m}x}{4}I\left(D_{1,i^{*}}^{(j)}<-\frac{\mathfrak{c}_{m}x}{4}\right).$

and

$\bar{\mathfrak{D}}_{2,i^{*}}^{(j)}\equiv\mathfrak{D}_{2,i^{*}}^{(j)}I\Bigg{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\leq\mathfrak{D}_{2,i^{*}}^{(j)}\leq
1\Bigg{)}+I\Bigg{(}\mathfrak{D}_{2,i^{*}}^{(j)}>1\Bigg{)}+\Big{(}\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Big{)}I\Bigg{(}\mathfrak{D}_{2,i^{*}}^{(j)}<\frac{9\mathfrak{c}_{m}^{2}}{16}-1\Bigg{)}.$

We will first prove the two bounds in (C.27) and (C.28) for $E_{1}$ and
$E_{2}$, which will use the following two properties:

######  Property C.4.

For any $i,j\in\\{1,\dots,n\\}$,

$\|\bar{D}_{1,i^{*},x}-\bar{D}_{1,i^{*},x}^{(j)}\|_{2}=\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(j)}\|_{2}\text{
and
}\|\bar{\mathfrak{D}}_{2,i^{*}}-\bar{\mathfrak{D}}_{2,i^{*}}^{(j)}\|_{3/2}=\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(j)}\|_{3/2}.$

######  Property C.5.

For any $i\in\\{1,\dots,n\\}$,

$\|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}\|_{2}\leq
2\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}\text{ and
}\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2}\leq
2\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}$

###### Proof of Properties C.4 and C.5.

Note that Property C.4 is true because $X_{1}^{*},\dots,X_{n}^{*}$ are
independent copies of $X_{1},\dots,X_{n}$, and Property C.5 is true because of
the triangular inequalities

$\|\bar{D}_{1,x}-\bar{D}_{1,i^{*},x}\|_{2}\leq\|\bar{D}_{1,x}-\bar{D}_{1,x}^{(i)}\|_{2}+\underbrace{\|\bar{D}_{1,x}^{(i)}-\bar{D}_{1,i^{*},x}\|_{2}}_{=\|\bar{D}_{1,x}^{(i)}-\bar{D}_{1,x}\|_{2}}$

and

$\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2}\leq\|\bar{\mathfrak{D}}_{2}-\bar{\mathfrak{D}}_{2}^{(i)}\|_{3/2}+\underbrace{\|\bar{\mathfrak{D}}_{2}^{(i)}-\bar{\mathfrak{D}}_{2,i^{*}}\|_{3/2}}_{\|\bar{\mathfrak{D}}_{2}^{(i)}-\bar{\mathfrak{D}}_{2}\|_{3/2}},$

as well as Property C.4. ∎

####  C.2.1. Proof of the bound for $E_{1}$, (C.27)

Recall

$E_{1}=\sum_{i=1}^{n}\mathbb{E}\left[\int_{-1}^{1}\left\\{f_{x}^{\prime}(W_{b}+\Delta)-f_{x}^{\prime}(W_{b}^{(i)}+\Delta_{i^{*}}+t)\right\\}k_{b,i}(t)dt\right].$