∎

11institutetext:  Lê Vǎn Thành 22institutetext: Department of Mathematics,
Vinh University, Nghe An, Vietnam
22email<EMAIL_ADDRESS>
# On a new concept of stochastic domination and the laws of large numbers

Lê Vǎn Thành

###### Abstract

Consider a sequence of positive integers $\\{k_{n},n\geq 1\\}$, and an array
of nonnegative real numbers $\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$
satisfying $\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}=C_{0}\in(0,\infty).$ This
paper introduces the concept of $\\{a_{n,i}\\}$-stochastic domination. We
develop some techniques concerning this concept and apply them to remove an
assumption in a strong law of large numbers of Chandra and Ghosal [Acta. Math.
Hungarica, 1996]. As a by-product, a considerable extension of a recent result
of Boukhari [J. Theoret. Probab., 2021] is established and proved by a
different method. The results on laws of large numbers are new even when the
summands are independent. Relationships between the concept of
$\\{a_{n,i}\\}$-stochastic domination and the concept of
$\\{a_{n,i}\\}$-uniform integrability are presented. Two open problems are
also discussed.

###### Keywords:

Stochastic domination Uniform integrability Strong law of large numbers Weak
law of large numbers Weighted sum Cesàro stochastic domination

###### MSC:

60E15 60F05 60F15

##  1 Introduction and Motivation

Let $1\leq p<2$. The classical Marcinkiewicz–Zygmund strong law of large
numbers (SLLN) states that for a sequence $\\{X_{n},n\geq 1\\}$ of independent
identically distributed mean zero random variables, condition
$\mathbb{E}(|X_{1}|^{p})<\infty$ is necessary and sufficient for

$\lim_{n\to\infty}\dfrac{\sum_{i=1}^{n}X_{i}}{n^{1/p}}=0\text{ almost surely
(a.s.).}$ (1.1)

Now, let us recall a weak dependence structure introduced in Chandra and
Ghosal chandra1996extensions  as follows. A sequence of random variables
$\\{X_{n},n\geq 1\\}$ is said to be asymptotically almost negatively
associated (AANA) if there exists a sequence of nonnegative real numbers
$\\{q_{n},n\geq 1\\}$ with $\lim_{n\to\infty}q_{n}=0$ such that

$\operatorname{Cov}(f(X_{n}),g(X_{n+1},\ldots,X_{n+k}))\leq
q_{n}\left(\operatorname{Var}(f(X_{n}))\operatorname{Var}(g(X_{n+1},\ldots,X_{n+k}))\right)^{1/2},$

for all $n\geq 1,k\geq 1$ and for all coordinatewise nondecreasing continuous
functions $f$ and $g$ provided the right side of the above inequality is
finite. The $q_{n},n\geq 1$ are called mixing coefficients. The starting point
of the current investigation is the following SLLN established by Chandra and
Ghosal chandra1996extensions .

######  Theorem 1.1 (Chandra and Ghosal chandra1996extensions )

Let $1\leq p<2$ and let $\\{X_{n},n\geq 1\\}$ be a sequence of AANA mean zero
random variables with the sequence of mixing coefficients satisfying
$\sum_{n=1}^{\infty}q_{n}^{2}<\infty$. Let

$G(x)=\sup_{n\geq 1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{i}|>x),\
x\in\mathbb{R}.$

If

$\int_{0}^{\infty}x^{p-1}G(x)\operatorname{d}x<\infty,$ (1.2)

and

$\sum_{n=1}^{\infty}\mathbb{P}(|X_{n}|^{p}>n)<\infty,$ (1.3)

then the Marcinkiewicz–Zygmund SLLN (1.1) is obtained.

The above result of Chandra and Ghosal chandra1996extensions  weakens the
assumptions in the classical Marcinkiewicz–Zygmund SLLN not only by
considering a weak dependence structure, but also by relaxing the identical
distribution condition. We refer to (1.2) and (1.3) as the Chandra–Ghosal
conditions. It is clear that for a sequence $\\{X_{n},n\geq 1\\}$ of random
variables with a common law, (1.2) and (1.3) are equivalent since each of them
is equivalent to $\mathbb{E}(|X_{1}|^{p})<\infty$. This leads to a natural
question in this context is whether (1.2) or (1.3) can be removed. The current
work is an attempt to answer this question. More precisely, we shall prove the
following theorem.

###### Theorem 1.2

Theorem 1.1 holds without Condition (1.3).

To prove Theorem 1.2, we develop some results concerning a new concept of
stochastic domination which leads to the concept of the Cesàro stochastic
domination as a particular case.

Let $\\{k_{n},n\geq 1\\}$ be a sequence of positive integers. An array
$\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ of random variables is said to be
stochastically dominated by a random variable $X$ if

$\sup_{1\leq i\leq k_{n},n\geq
1}\mathbb{P}(|X_{n,i}|>x)\leq\mathbb{P}(|X|>x),\ \text{ for all
}x\in\mathbb{R}.$ (1.4)

This concept was extended to the concept of the so-called Cesàro stochastic
domination by Gut gut1992complete  as follows. An array $\\{X_{n,i},1\leq
i\leq k_{n},n\geq 1\\}$ of random variables is said to be stochastically
dominated in the Cesàro sense (or weakly mean dominated) by a random variable
$X$ if

$\sup_{n\geq 1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{P}(|X_{n,i}|>x)\leq
C\mathbb{P}(|X|>x),\ \text{ for all }x\in\mathbb{R},$ (1.5)

where $C>0$ is a constant. It was shown by Gut (gut1992complete, , Example
2.1) that (1.5) is strictly weaker than (1.4).

We will now introduce a new concept of stochastic domination. Let
$\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of nonnegative real
numbers satisfying

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}=C_{0}\in(0,\infty).$ (1.6)

An array $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ of random variables is said
to be $\\{a_{n,i}\\}$-stochastically dominated by a random variable $X$ if

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\leq
C_{0}\mathbb{P}(|X|>x),\ \text{ for all }x\in\mathbb{R}.$ (1.7)

In view of Gut’s definition in (1.5), one may be tempted to give an apparently
weaker definition of $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ being
$\\{a_{n,i}\\}$-stochastically dominated by a random variable $Y$, namely that

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\leq
C\mathbb{P}(|Y|>x),\ \text{ for all }x\in\mathbb{R},$ (1.8)

for some finite constant $C>0$. However, it will be shown in Theorem 2.2 that
(1.7) and (1.8) are indeed equivalent. Therefore, concerning Gut’s definition
of the Cesàro stochastic domination, we can simply choose $C=1$ in (1.5). If
$a_{n,i}=1/k_{n},1\leq i\leq k_{n},\ n\geq 1,$ then it is obvious that
$C_{0}=1$, and the concept of $\\{a_{n,i}\\}$-stochastic domination reduces to
the concept of stochastic domination in the Cesàro sense.

If $0<p<1$ and $\\{X_{n},n\geq 1\\}$ is a sequence of random variables
satisfying (1.2) and (1.3), then the Marcinkiewicz–Zygmund SLLN (1.1) is valid
irrespective of any dependence structure (see Remark 3 in Chandra and Ghosal
chandra1996extensions ). Boukhari boukhari2021weak  recently used techniques
from martingales theory to prove that a similar result holds true for the weak
law of large numbers (WLLN) for maximal partial sums with general normalizing
sequences. The tools developed in this paper also allow us to establish an
extension of Theorem 1.2 of Boukhari boukhari2021weak . A special case of our
WLLNs in Section 4 is the following theorem.

###### Theorem 1.3

Let $\\{X_{n},n\geq 1\\}$ be a sequence of random variables, $G(\cdot)$ as in
Theorem 1.1 and let $\\{b_{n},n\geq 1\\}$ be a nondecreasing sequence of
positive real numbers such that

$\sum_{i=1}^{n}\dfrac{b_{i}}{i^{2}}=O\left(\dfrac{b_{n}}{n}\right).$ (1.9)

If

$\lim_{k\to\infty}kG(b_{k})=0,$ (1.10)

then the WLLN

$\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}X_{i}\right|\overset{\mathbb{P}}{\to}0\text{ as
}n\to\infty$ (1.11)

is obtained.

###### Remark 1.4

Boukhari (boukhari2021weak, , Theorem 1.2) proved Theorem 1.3 under a stronger
condition that the sequence $\\{X_{n},n\geq 1\\}$ is stochastically dominated
by a random variable $X$ satisfying

$\lim_{k\to\infty}k\mathbb{P}(|X|>b_{k})=0.$

An example in Section 4 shows that for $0<p<1$ and $b_{n}=n^{1/p}$, there
exists a sequence of random variables $\\{X_{n},n\geq 1\\}$ with no
stochastically dominating random variable, but (1.10) is satisfied and
therefore the WLLN (1.11) is valid. Our proof of Theorem 1.3 is simpler than
that of Theorem 1.2 of Boukhari boukhari2021weak  in the sense that we do not
use the Doob maximal inequality for martingales as was done in Boukhari
boukhari2021weak . The WLLN for dependent random variables and random vectors
was also studied in hien2015weak ; kruglov2011generalization ;
rosalsky2009weak , among others.

The rest of the paper is organized as follows. In Section 2, we prove the
equivalence between the definitions of $\\{a_{n,i}\\}$-stochastic domination
given in (1.7) and (1.8). It is also shown that certain bounded moment
conditions on an array of random variables $\\{X_{n,i},1\leq i\leq k_{n},n\geq
1\\}$ can accomplish the concept of $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$
being $\\{a_{n,i}\\}$-stochastically dominated. Section 3 discusses about
relationships between the concept of $\\{a_{n,i}\\}$-stochastic domination and
the concept of $\\{a_{n,i}\\}$-uniform integrability. Strong and weak laws of
large numbers for triangular arrays of random variables are presented in
Section 4. From these general results, Theorems 1.2 and 1.3 follow. Section 5
contains further remarks and two open problems.

Notation: Throughout this paper, $\\{k_{n},n\geq 1\\}$ is assumed to be a
sequence of positive integers. For a set $A$, $\mathbf{1}(A)$ denotes the
indicator function of $A$. For $x\geq 0$, let $\log x$ denote the logarithm
base $2$ of $\max\\{2,x\\}$. For $x\geq 0$ and for a fixed positive integer
$\nu$, let

$\log_{\nu}(x):=(\log x)(\log\log x)\ldots(\log\cdots\log x),$ (1.12)

and

$\log_{\nu}^{(2)}(x):=(\log x)(\log\log x)\ldots(\log\cdots\log x)^{2},$
(1.13)

where in both (1.12) and (1.13), there are $\nu$ factors. For example,
$\log_{2}(x)=(\log x)(\log\log x)$, $\log_{3}^{(2)}(x)=(\log x)(\log\log
x)(\log\log\log x)^{2}$, and so on.

##  2 On the concept of $\\{a_{n,i}\\}$-stochastic domination

In this section, we employ some properties of slowly varying functions as well
as techniques in Rosalsky and Thành rosalsky2021note  to prove some results on
the concept of $\\{a_{n,i}\\}$-stochastic domination. We note that all results
in Sections 2 and 3 are stated for a triangular array $\\{X_{n,i},1\leq i\leq
k_{n},n\geq 1\\}$ of random variables but they still hold for a sequence of
random variables $\\{X_{n},n\geq 1\\}$ by considering $X_{n,i}=X_{i}$, $1\leq
i\leq k_{n},n\geq 1$.

The following theorem is a simple result and its proof is similar to that of
Theorem 2.1 of Rosalsky and Thành rosalsky2021note . It plays a useful role in
proving the laws of large numbers in Section 4.

###### Theorem 2.1

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables,
let $\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of nonnegative real
numbers satisfying (1.6) and let

$F(x)=1-\dfrac{1}{C_{0}}\sup\limits_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}.$

Then $F(\cdot)$ is the distribution function of a random variable $X$ if and
only if $\lim_{x\to\infty}F(x)=1.$ In such a case, $\\{X_{n,i},1\leq i\leq
k_{n},n\geq 1\\}$ is $\\{a_{n,i}\\}$-stochastically dominated by $X$.

###### Proof

It is clear that $F(\cdot)$ is nondecreasing, and

$\lim\limits_{x\to-\infty}F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}=0.$

Let $\varepsilon>0$ be arbitrary. For $a\in\mathbb{R}$, let $n_{0}\geq 1$ be
such that

$\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>a)>\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>a)-\varepsilon/2,$

or equivalently,

$1-\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>a)<F(a)+\varepsilon/2.$
(2.1)

Since the function

$x\mapsto\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>x),\
x\in\mathbb{R},$

is nonincreasing and right continuous, there exists $\delta>0$ such that

$-\varepsilon/2<\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>x)-\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>a)\leq
0\text{ for all $x$ such that }0\leq x-a<\delta.$

Therefore, for $x$ satisfying $0\leq x-a<\delta$, we have

$\displaystyle F(x)-\varepsilon$
$\displaystyle=1-\dfrac{1}{C_{0}}\sup\limits_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)-\varepsilon$
$\displaystyle\leq
1-\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>x)-\varepsilon$
$\displaystyle<1-\dfrac{1}{C_{0}}\sum_{i=1}^{k_{n_{0}}}a_{n_{0},i}\mathbb{P}(|X_{n_{0},i}|>a)-\varepsilon/2$
$\displaystyle<F(a)\ \text{ (by \eqref{df01})}$

and so $|F(x)-F(a)|<\varepsilon$. Thus $\lim\limits_{x\to a^{+}}F(x)=F(a)$.
Since $a\in\mathbb{R}$ is arbitrary, this implies that $F$ is right continuous
on $\mathbb{R}$. Since $F(\cdot)$ is nondecreasing, right continuous and
$\lim\limits_{x\to-\infty}F(x)=0$, it is the distribution function of a random
variable $X$ if and only if $\lim\limits_{x\to\infty}F(x)=1$. By definition of
$F(\cdot)$, $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by $X$. ∎

The next theorem establishes the equivalence between the definitions of
$\\{a_{n,i}\\}$-stochastic domination given in (1.7) and (1.8). A similar
result concerning the concept of stochastic domination was proved by Rosalsky
and Thành rosalsky2021note .

###### Theorem 2.2

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables
and let $\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of nonnegative
real numbers satisfying (1.6). Then there exists a random variable $X$
satisfying (1.7) if and only if there exist a random variable $Y$ and a finite
constant $C>0$ satisfying (1.8).

Moreover, (i) if $g:[0,\infty)\to[0,\infty)$ is a measurable function with
$g(0)=0$ which is bounded on $[0,A]$ and differentiable on $[A,\infty)$ for
some $A\geq 0$ or (ii) if $g:[0,\infty)\to[0,\infty)$ is a continuous function
which is eventually nondecreasing with $\lim_{x\to\infty}g(x)=\infty$, then
the condition $\mathbb{E}(g(|Y|))<\infty$ where $Y$ is as in (1.8) implies
that $\mathbb{E}(g(|X|))<\infty$ where $X$ is as in (1.7).

###### Proof

The necessity half is immediate by taking $Y=X$ and $C=C_{0}$. Conversely, if
there exist a nonnegative random variable $Y$ and a finite constant $C>0$
satisfying (1.8), then

$\begin{split}\lim_{x\to\infty}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\leq
C\lim_{x\to\infty}\mathbb{P}(|Y|>x)=0,\end{split}$

and so by Theorem 2.1, there exists a random variable $X$ with distribution
function

$F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}.$

This implies

$\begin{split}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)=C_{0}(1-F(x))=C_{0}\mathbb{P}(X>x),\
x\in\mathbb{R}\end{split}$

thereby verifying (1.7).

The rest of the proof proceeds in a similar manner as that of Theorem 2.4 (i)
and (ii) in Rosalsky and Thành rosalsky2021note . The details will be omitted.
∎

The following result is a direct consequence of Theorem 2.2 by choosing
$a_{n,i}=1/k_{n}$ for all $1\leq i\leq k_{n}$, $n\geq 1$. It says that in the
Cesàro stochastic domination definition, (1.5) can be simplified to

$\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{P}(|X_{n,i}|>x)\leq\mathbb{P}(|Y|>x),\
\text{ for all }x\in\mathbb{R},$ (2.2)

for some random variable $Y$ (surprisingly, this was not noticed by Gut
gut1992complete ).

###### Corollary 2.3

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables.
Then there exist a random variable $X$ and a finite constant $C>0$ satisfying
(1.5) if and only if there exists a random variable $Y$ satisfying (2.2).
Moreover, if $g$ is a measurable function satisfying assumptions in Theorem
2.2, then the condition $\mathbb{E}(g(|X|))<\infty$ where $X$ is as in (1.5)
implies that $\mathbb{E}(g(|Y|))<\infty$ where $Y$ is as in (2.2).

By using integration by parts, we have from (2.2) that for all $r>0$ and
$x\geq 0$

$\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{E}\left(|X_{n,i}|^{r}\mathbf{1}(|X_{n,i}|\leq
x)\right)\leq\mathbb{E}\left(|Y|^{r}\mathbf{1}(|Y|\leq
x)\right)+x^{r}\mathbb{P}(|Y|>x),$ (2.3)

and

$\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{E}\left(|X_{n,i}|^{r}\mathbf{1}(|X_{n,i}|>x)\right)\leq\mathbb{E}\left(|Y|^{r}\mathbf{1}(|Y|>x)\right).$
(2.4)

We will use (2.3) and (2.4) in our proofs without further mention.

The following consequence of Theorem 2.1 is also useful in proving the laws of
large numbers.

###### Corollary 2.4

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables
and let

$F(x)=1-\sup\limits_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}.$

Then $F$ is the distribution function of a random variable $X$ if and only if
$\lim_{x\to\infty}F(x)=1.$ In such a case, $\\{X_{n,i},1\leq i\leq k_{n},n\geq
1\\}$ is stochastically dominated in the Cesàro sense by $X$.

In view of (2.2), if $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
stochastically dominated in the Cesàro sense by a random variable $Y$, then it
is $\\{a_{n,i}\\}$-stochastically dominated by $Y$ with $a_{n,i}=1/k_{n}$,
$1\leq i\leq k_{n}$, $n\geq 1$. The following example shows that the concept
of $\\{a_{n,i}\\}$-stochastic domination is strictly weaker than the concept
of stochastic domination in the Cesàro sense.

###### Example 2.5

Let $k_{n}\equiv n$ and let $m_{n}$ be the greatest integer number which is
less than or equal to $n/2$. Let $a_{n,i}=1/m_{n}$ for $1\leq i\leq m_{n},\
n\geq 2$ and $a_{n,i}=1/n^{2}$ for $m_{n}<i\leq n,n\geq 1$. Let
$\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of random variables such
that

$\mathbb{P}(X_{n,i}=-1)=\mathbb{P}(X_{n,i}=1)=1/2,\ 1\leq i\leq m_{n},\ n\geq
2$

and

$\mathbb{P}(X_{n,i}=-n)=\mathbb{P}(X_{n,i}=n)=1/2,\ m_{n}<i\leq n,\ n\geq 1.$

Then, for $x\geq 1$, we have

$\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x)=\begin{cases}0&\text{ if
}n\leq x,\\\ \dfrac{n-m_{n}}{n}&\text{ if }n>x.\end{cases}$

This implies

$\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x)\geq\dfrac{1}{2},\ \text{
for all }x\geq 1.$

Thus by Corollary 2.4, there is no random variable $Y$ such that
$\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is stochastically dominated in the
Cesàro sense by $Y$. Now, we have

$1<C_{0}=\sup_{n\geq 1}\sum_{i=1}^{n}a_{n,i}=\sup_{n\geq
2}\left(1+\dfrac{n-m_{n}}{n^{2}}\right)\leq 2,$

and for $x\geq 1$ and $n\geq 1$,

$\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x)=\sum_{i=m_{n}+1}^{n}\dfrac{1}{n^{2}}\mathbb{P}(|X_{n,i}|>x)=\begin{cases}0&\text{
if }n\leq x,\\\ \dfrac{n-m_{n}}{n^{2}}\leq\dfrac{1}{n}&\text{ if
}n>x.\end{cases}$

Thus $\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\to 0$ as $x\to\infty$. By
Theorem 2.1, $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by a random variable $X$ with
distribution function

$F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}.$

This shows that the concept of $\\{a_{n,i}\\}$-stochastic domination is
strictly weaker than the concept of stochastic domination in the Cesàro sense.

Recall that a real-valued function $L(\cdot)$ is said to be slowly varying (at
infinity) if it is a positive and measurable function on $[A,\infty)$ for some
$A\geq 0$, and for each $\lambda>0$,

$\lim_{x\to\infty}\dfrac{L(\lambda x)}{L(x)}=1.$

If $L(\cdot)$ is a slowly varying function, then there exists a slowly varying
function $\tilde{L}(\cdot)$, unique up to an asymptotic equivalence,
satisfying

$\lim_{x\to\infty}L(x)\tilde{L}\left(xL(x)\right)=1\ \text{ and
}\lim_{x\to\infty}\tilde{L}(x)L\left(x\tilde{L}(x)\right)=1.$ (2.5)

The function $\tilde{L}(\cdot)$ is called the de Bruijn conjugate of
$L(\cdot)$ (see Bingham et al. (bingham1989regular, , p. 29)). For many “nice”
slowly varying functions $L(\cdot)$, we can choose $\tilde{L}(x)=1/L(x)$.
Especially, if $L(x)=(\log x)^{\gamma}$ or $L(x)=(\log\log x)^{\gamma}$ for
some $\gamma\in\mathbb{R}$, then $\tilde{L}(x)=1/L(x)$.

Let $L(\cdot)$ be a slowly varying function and let $\alpha>0$. By using a
suitable asymptotic equivalence version (see Lemma 2.2 and Lemma 2.3 (i) in
Anh et al. anh2021marcinkiewicz ), we can firstly assume that $L(\cdot)$ is
positive and differentiable on $[a,\infty)$, and $x^{\alpha}L(x)$ is strictly
increasing on $[a,\infty)$ for some large $a$. Next, let $L_{1}(\cdot)$ be a
slowly varying function satisfying $L_{1}(x)=L(a)x/a$ if $0\leq x<a$ and
$L_{1}(x)=L(x)$ if $x\geq a$ (i.e., $L_{1}(0)=0$ with a linear growth to
$L(a)$ over $[0,a)$, and $L_{1}(x)\equiv L(x)$ on $[a,\infty)$). Then (i)
$L_{1}(x)$ is continuous on $[0,\infty)$ and differentiable on $[a,\infty)$,
and (ii) $x^{\alpha}L_{1}(x)$ is strictly increasing on $[0,\infty)$. In this
paper, we will assume, without loss of generality, that these properties are
fulfilled for the underlying slowly varying functions.

The next theorem shows that bounded moment conditions on an array of random
variables $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ with respect to weights
$\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ can accomplish
$\\{a_{n,i}\\}$-stochastic domination.

###### Theorem 2.6

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables,
$\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ an array of nonnegative real numbers
satisfying (1.6). Let $p>0$ and let $\nu$ be a fixed positive integer. Let
$L(\cdot)$ be a slowly varying function. If

$\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}\left(|X_{n,i}|^{p}L(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\right)<\infty,$
(2.6)

then there exists a random variable $X$ with distribution function

$F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}$ (2.7)

such that $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by $X$, and

$\mathbb{E}\left(|X|^{p}L(|X|)\right)<\infty.$ (2.8)

###### Remark 2.7

A weaker version of Theorem 2.6 for stochastic domination (without the
appearance of the slowly varying function $L(\cdot)$) was proved by Rosalsky
and Thành rosalsky2021note  (see Theorem 2.5 (ii) and (iii) in Rosalsky and
Thành rosalsky2021note ). Typical examples of slowly varying functions
$L(\cdot)$ for (2.6) are $L(x)\equiv 1$ and $L(x)\equiv
L_{1}(x)(\log_{\nu}^{(2)}(x))^{-1}$, where $L_{1}(\cdot)$ is another slowly
varying function. Theorem 2.6 is proved by employing an idea from Galambos and
Seneta galambos1973regularly .

Before proving Theorem 2.6, we recall a simple result on the expectation of a
nonnegative random variable, see Rosalsky and Thành rosalsky2021note  for a
proof.

###### Lemma 2.8

Let $h:[0,\infty)\to[0,\infty)$ be a measurable function with $h(0)=0$ which
is bounded on $[0,A]$ and differentiable on $[A,\infty)$ for some $A\geq 0$.
If $\xi$ is a nonnegative random variable, then

$\begin{split}\mathbb{E}(h(\xi))&=\mathbb{E}(h(\xi)\mathbf{1}(\xi\leq
A))+h(A)+\int_{A}^{\infty}h^{\prime}(x)\mathbb{P}(\xi>x)\operatorname{d}x.\end{split}$

Proof of Theorem 2.6. Set

$g(x)=x^{p}L(x)\log_{\nu}^{(2)}(x),\ \text{ and }h(x)=x^{p}L(x),\ x\geq 0.$

Since $\lim_{x\to\infty}g(x)=\infty$ and $g(\cdot)$ is strictly increasing on
$[0,\infty)$ as we have assumed before, we have from Markov’s inequality and
(2.6) that

$0\leq\lim_{x\to\infty}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\leq\lim_{x\to\infty}\dfrac{1}{g(x)}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(g(|X_{n,i}|))=0.$

By Theorem 2.1, the array $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by a random variable $X$ with
distribution function $F(\cdot)$ given in (2.7).

Next, we prove (2.8). We firstly consider the case where the slowly varying
function $L(\cdot)$ is differentiable on an infinite interval far enough from
$0$, and

$\lim_{x\to\infty}\dfrac{xL^{\prime}(x)}{L(x)}=0.$ (2.9)

By (2.9), there exists $B\geq 0$ such that

$\left|\dfrac{xL^{\prime}(x)}{L(x)}\right|\leq\dfrac{p}{2},\ x>B.$

It follows that

$h^{\prime}(x)=px^{p-1}L(x)+x^{p}L^{\prime}(x)=x^{p-1}L(x)\left(p+\dfrac{xL^{\prime}(x)}{L(x)}\right)\leq\dfrac{3px^{p-1}L(x)}{2},\
x\geq B.$ (2.10)

Therefore, there exists a constant $C_{1}$ such that

$\begin{split}\mathbb{E}(h(X))&=\mathbb{E}(h(X)\mathbf{1}(X\leq
B))+h(B)+\int_{B}^{\infty}h^{\prime}(x)\mathbb{P}(X>x)\mathrm{d}x\\\ &\leq
C_{1}+\dfrac{3p}{2}\int_{B}^{\infty}x^{p-1}L(x)\mathbb{P}(X>x)\mathrm{d}x\\\
&=C_{1}+\dfrac{3p}{2C_{0}}\int_{B}^{\infty}x^{p-1}L(x)\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x)\mathrm{d}x\\\ &\leq
C_{1}+\dfrac{3p}{2C_{0}}\int_{B}^{\infty}\dfrac{1}{x\log_{\nu}^{(2)}(x)}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}\left(g(|X_{n,i}|)\right)\mathrm{d}x\\\
&=C_{1}+\dfrac{3p}{2C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}\left(g(|X_{n,i}|)\right)\int_{B}^{\infty}\dfrac{\mathrm{d}x}{x\log_{\nu}^{(2)}(x)}<\infty,\end{split}$

where we have applied Lemma 2.8 in the first equality, (2.10) in the first
inequality, Markov’s inequality in the second inequality, and (2.6) in the
last inequality. Thus we obtain (2.8) in this case.

For general slowly varying function $L(\cdot)$, by a result from page 111 of
Galambos and Seneta galambos1973regularly , there exists a slowly varying
function $L_{1}(\cdot)$ which is differentiable on $[B_{1},\infty)$ for some
$B_{1}$ large enough, and satisfies

$\lim_{x\to\infty}\dfrac{L_{1}(x)}{L(x)}=1$ (2.11)

and

$\lim_{x\to\infty}\dfrac{xL_{1}^{\prime}(x)}{L_{1}(x)}=0.$

For $n\geq 1$, $1\leq i\leq n$, we have from (2.11) that for all $B_{2}$ large
enough

$\begin{split}&\mathbb{E}\left(|X_{n,i}|^{p}L_{1}(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\right)\\\
&\qquad=\mathbb{E}\left(|X_{n,i}|^{p}L_{1}(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\mathbf{1}(|X_{n,i}|\leq
B_{2})\right)\\\
&\quad\qquad+\mathbb{E}\left(|X_{n,i}|^{p}L_{1}(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\mathbf{1}(|X_{n,i}|>B_{2})\right)\\\
&\qquad\leq
C_{2}+2\mathbb{E}\left(|X_{n,i}|^{p}L(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\mathbf{1}(|X_{n,i}|>B_{2})\right),\end{split}$
(2.12)

where $C_{2}$ is a finite constant. Combining (2.6) and (2.12) yields

$\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}\left(|X_{n,i}|^{p}L_{1}(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\right)<\infty.$
(2.13)

Proceeding exactly the same manner as the first case with $L(\cdot)$ is
replaced by $L_{1}(\cdot)$, we obtain (2.8). The proof of the theorem is
completed. ∎

The following corollary follows immediately from Theorem 2.6.

###### Corollary 2.9

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables
and let $L(\cdot)$ be a slowly varying function. Let $p>0$ and let $\nu$ be a
fixed positive integer. If

$\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{E}\left(|X_{n,i}|^{p}L(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\right)<\infty,$

then there exists a random variable $X$ with distribution function

$F(x)=1-\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R}$

such that $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is stochastically
dominated in the Cesàro sense by $X$, and

$\mathbb{E}\left(|X|^{p}L(|X|)\right)<\infty.$

##  3 Relationships between $\\{a_{n,i}\\}$-stochastic domination and
$\\{a_{n,i}\\}$-uniform integrability

The concept of $\\{a_{n,i}\\}$-uniform integrability was introduced by Ordóñez
Cabrera ordonezcabrera1994convergence . Let $\\{a_{n,i},1\leq i\leq
k_{n},n\geq 1\\}$ be an array of nonnegative real numbers satisfying (1.6). An
array $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ of random variables is said to
be $\\{a_{n,i}\\}$-uniformly integrable if

$\lim_{a\to\infty}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(|X_{n,i}|\mathbf{1}(|X_{n,i}|>a))=0.$

Similar to the classical characterization of the uniform integrability, it was
proved by Ordóñez Cabrera ordonezcabrera1994convergence  that an array of
random variables $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-uniformly integrable if and only if

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(|X_{n,i}|)<\infty$

and for each $\varepsilon>0$, there exists $\delta>0$ such that whenever
$\\{A_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is an array of events satisfying

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(A_{n,i})<\delta,$

then

$\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(|X_{n,i}|\mathbf{1}(A_{n,i}))<\varepsilon.$

If $a_{n,i}=1/k_{n},1\leq i\leq k_{n},\ n\geq 1,$ then it reduces to the
concept of $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ being uniformly
integrable in the Cesàro sense which was introduced in chandra1989uniform .
The de La Vallée–Poussin criterion for uniform integrability in the Cesàro
sense, and for $\\{a_{n,i}\\}$-uniform integrability was proved, respectively,
by Chandra and Goswami chandra1992cesaro , and Ordóñez Cabrera
ordonezcabrera1994convergence . The former is a special case of the latter,
which reads as follows: An array of random variables $\\{X_{n,i},1\leq i\leq
k_{n},n\geq 1\\}$ is $\\{a_{n,i}\\}$-uniformly integrable if and only if there
exists a measurable function $g:[0,\infty)\to[0,\infty)$ with $g(0)=0$,
$g(x)/x\to\infty$ as $x\to\infty$, and

$\sup_{n\geq 1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(g(|X_{n,i}|))<\infty.$

Moreover, $g$ can be selected to be convex and such that $g(x)/x$ is
nondecreasing.

The next theorem establishes relationships between the concept of
$\\{a_{n,i}\\}$-stochastic domination and the concept of
$\\{a_{n,i}\\}$-uniform integrability.

###### Theorem 3.1

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables
and let $\\{a_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of nonnegative
real numbers satisfying (1.6). Let $p>0$ and let $\tilde{L}(\cdot)$ be the
Brujin conjugate of a slowly varying function $L(\cdot)$.

(i)

If $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is $\\{a_{n,i}\\}$-stochastically
dominated by a random variable $X$ with
$\mathbb{E}(|X|^{p}L(|X|^{p}))<\infty$, then
$\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-uniformly integrable.

(ii)

If $\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-uniformly integrable, then there exists a random variable $X$
with distribution function

$F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x),x\in\mathbb{R},$

such that $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by $X$,

$\mathbb{E}\left(|X|^{p}L(|X|^{p})(\log_{\nu}^{(2)}(|X|))^{-1}\right)<\infty\text{
for all fixed positive integer }\nu,$ (3.1)

and

$\lim_{x\to\infty}x\mathbb{P}\left(|X|>x^{1/p}\tilde{L}^{1/p}(x)\right)=0.$
(3.2)

###### Proof

Let $f(x)=x^{p}L(x^{p}),\ g(x)=x^{1/p}\tilde{L}^{1/p}(x),\ x\geq 0$. Recalling
that we assume, without loss of generality, that $f$ and $g$ are strictly
increasing on $[0,\infty)$.

(i) Since $\mathbb{E}(|X|^{p}L(|X|^{p}))<\infty$, it follows from the
classical de La Vallée Poussin criterion for uniform integrability that there
exists a continuous and strictly increasing function
$h:[0,\infty)\to[0,\infty)$ with $h(0)=0$, $\lim_{x\to\infty}h(x)/x=\infty$,
and $\mathbb{E}(h(|X|^{p}L(|X|^{p})))<\infty.$ Since $f(x)$ is strictly
increasing on $[0,\infty)$, the $\\{a_{n,i}\\}$-stochastic domination
assumption ensures that for all $n\geq 1$,

$\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>f^{-1}(h^{-1}(x)))\leq
C_{0}\mathbb{P}(|X|>f^{-1}(h^{-1}(x))),\ x\in\mathbb{R}$

or, equivalently,

$\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|^{p}L(|X_{n,i}|^{p})>h^{-1}(x))\leq
C_{0}\mathbb{P}(|X|^{p}L(|X|^{p})>h^{-1}(x)),\ x\in\mathbb{R}$

which, in turn, is equivalent to

$\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(h(|X_{n,i}|^{p}L(|X_{n,i}|^{p}))>x)\leq
C_{0}\mathbb{P}(h(|X|^{p}L(|X|^{p}))>x),\ x\in\mathbb{R}.$

It follows that

$\begin{split}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(h(|X_{n,i}|^{p}L(|X_{n,i}|^{p})))&=\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\int_{0}^{\infty}\mathbb{P}(h(|X_{n,i}|^{p}L(|X_{n,i}|^{p}))>x)\operatorname{d}x\\\
&\leq
C_{0}\int_{0}^{\infty}\mathbb{P}(h(|X|^{p}L(|X|^{p}))>x)\operatorname{d}x\\\
&=C_{0}\mathbb{E}(h(|X|^{p}L(|X|^{p})))<\infty.\end{split}$

By the de La Vallée Poussin criterion for $\\{a_{n,i}\\}$-uniform
integrability (Ordóñez Cabrera ordonezcabrera1994convergence ),
$\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-uniformly integrable.

(ii) Since $\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-uniformly integrable,

$\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}\left(|X_{n,i}|^{p}L(|X_{n,i}|^{p})\right)<\infty,$

and so by Theorem 2.6, $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by a random variable $X$ with
distribution function

$F(x)=1-\dfrac{1}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\ x\in\mathbb{R},$

and (3.1) holds.

Finally, by using the de La Vallée Poussin criterion for
$\\{a_{n,i}\\}$-uniform integrability again, there exists a nondecreasing
function $h$ defined on $[0,\infty)$ with $h(0)=0$ such that

$\lim_{x\to\infty}\dfrac{h(x)}{x}=\infty,$ (3.3)

and

$\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(h(|X_{n,i}|^{p}L(|X_{n,i}|^{p})))<\infty.$
(3.4)

By applying Lemma 2.1 in Anh et al. anh2021marcinkiewicz , we have
$f(g(x))/x\to 1$ as $x\to\infty$, and therefore

$f(g(x))>x/2\text{ for all large }x.$ (3.5)

We thus have from (3.3), (3.4), (3.5) and Markov’s inequality that

$\begin{split}\lim_{x\to\infty}x\mathbb{P}\left(|X|>g(x)\right)&=\dfrac{1}{C_{0}}\lim_{x\to\infty}x\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(|X_{n,i}|>g(x))\\\
&\leq\dfrac{1}{C_{0}}\lim_{x\to\infty}x\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(f(|X_{n,i}|)\geq f(g(x)))\\\
&\leq\dfrac{1}{C_{0}}\lim_{x\to\infty}x\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(f(|X_{n,i}|)\geq x/2)\\\
&\leq\dfrac{1}{C_{0}}\lim_{x\to\infty}x\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{P}(h(f(|X_{n,i}|))\geq h(x/2))\\\
&\leq\dfrac{1}{C_{0}}\lim_{x\to\infty}x\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\dfrac{\mathbb{E}(h(f(|X_{n,i}|)))}{h(x/2)}\\\
&=\dfrac{2}{C_{0}}\sup_{n\geq
1}\sum_{i=1}^{k_{n}}a_{n,i}\mathbb{E}(h(f(|X_{n,i}|)))\lim_{x\to\infty}\dfrac{x/2}{h(x/2)}=0,\end{split}$

thereby proving (3.2). ∎

The following corollary is a direct consequence of Theorem 3.1. It plays an
important role in establishing the weak laws of large numbers with general
normalizing sequences under the Cesàro uniform integrability condition in
Section 4.

###### Corollary 3.2

Let $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ be an array of random variables.
Let $p>0$ and let $L(\cdot)$ be a slowly varying function.

(i)

If $\\{X_{n,i},1\leq i\leq k_{n},n\geq 1\\}$ is stochastically dominated in
the Cesàro sense by a random variable $X$ with
$\mathbb{E}(|X|^{p}L(|X|^{p}))<\infty$, then
$\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is uniformly
integrable in the Cesàro sense.

(ii)

If $\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq k_{n},n\geq 1\\}$ is
uniformly integrable in the Cesàro sense, then there exists a random variable
$X$ with distribution function

$F(x)=1-\sup_{n\geq
1}\dfrac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbb{P}(|X_{n,i}|>x),x\in\mathbb{R},$

such that $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is stochastically dominated in
the Cesàro sense by $X$, and (3.1) and (3.2) hold.

##  4 Laws of large numbers for triangular arrays and proofs of Theorems 1.2
and 1.3

In this section, we establish strong and weak laws of large numbers for
triangular arrays of random variables. We say that a collection
$\\{X_{i},1\leq i\leq N\\}$ of random variables satisfies condition $(H)$ if
for all $a>0$, there exists a constant $C$ such that

$\mathbb{E}\left(\max_{1\leq k\leq
n}\left|\sum_{i=m+1}^{m+k}\left(X_{i}^{(a)}-\mathbb{E}X_{i}^{(a)}\right)\right|\right)^{2}\leq
C\sum_{i=m+1}^{m+n}\mathbb{E}(X_{i}^{(a)})^{2},\ m\geq 0,n\geq 1,m+n\leq N,$
(4.1)

where

$X_{i}^{(a)}=-a\mathbf{1}(X_{i}<-a)+X_{i}\mathbf{1}(|X_{i}|\leq
a)+a\mathbf{1}(X_{i}>a),\ 1\leq i\leq N.$

An infinite sequence of random variables $\\{X_{i},i\geq 1\\}$ is said to
satisfy condition $(H)$ if every finite subsequence satisfies condition $(H)$.
Many dependence structures meet this condition. For example, condition $(H)$
holds for negatively associated sequences, negatively superadditive dependent
sequences, AANA sequences with the sequence of mixing coefficients is in
$\ell_{2}$ (the mixing coefficients $q_{n},n\geq 1$ satisfying
$\sum_{n=1}^{\infty}q_{n}^{2}<\infty$). In Adler and Matuła adler2018exact ,
the authors used a similar condition to establish exact SLLNs (see Theorems
3.2 and 4.1 in adler2018exact ).

Throughout this section, the symbol $C$ denotes a positive universal constant
which is not necessarily the same in each appearance. We shall let the indices
$k_{n}$ in the previous sections be $k_{n}\equiv n$.

The following theorem establishes the rate of convergence in SLLN for maximal
partial sums from triangular arrays of dependent random variables under the
Chandra–Ghosal-type condition (see Condition (4.2) below).

###### Theorem 4.1

Let $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of mean zero random
variables such that for each $n\geq 1$ fixed, the collection $\\{X_{n,i},1\leq
i\leq n\\}$ satisfies condition $(H)$, and let

$G(x)=\sup_{n\geq 1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x),\
x\in\mathbb{R}.$

Let $L(\cdot)$ be a slowly varying function and let $1\leq p<2$. When $p=1$,
we further assume that $L(\cdot)$ is nondecreasing and $L(x)\geq 1$ for all
$x\geq 0$. If

$\int_{0}^{\infty}x^{p-1}L^{p}(x)G(x)\operatorname{d}x<\infty,$ (4.2)

then

$\sum_{n=1}^{\infty}n^{-1}\mathbb{P}\left(\max_{1\leq k\leq
n}\left|\sum_{i=1}^{k}X_{n,i}\right|>\varepsilon
n^{1/p}\tilde{L}(n^{1/p})\right)<\infty\text{ for all }\varepsilon>0,$ (4.3)

where $\tilde{L}(\cdot)$ is the Bruijn conjugate of $L(\cdot)$.

###### Proof

Since $L(\cdot)$ is a slowly varying function and $L(x)\geq 1$ for all $x\geq
0$ when $p=1$, it follows from (4.2) that $\lim_{x\to\infty}G(x)=0$. By
Corollary 2.4, the array $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is
stochastically dominated in the Cesàro sense by a random variable $X$ with
distribution function $F(x)=1-G(x),\ x\in\mathbb{R}.$ Thus

$G(x)=\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x)=\mathbb{P}(|X|>x),\
x\in\mathbb{R}.$ (4.4)

Using the same arguments as in the proof of Theorem 2.6, we can assume,
without loss of generality, that the function $L(\cdot)$ satisfies

$\lim_{x\to\infty}\dfrac{xL^{\prime}(x)}{L(x)}=0.$ (4.5)

Let $h(x)=x^{p}L^{p}(x),x\geq 0$. Then it follows from (4.5) that for all
large $x$,

$h^{\prime}(x)=px^{p-1}L^{p}(x)\left(1+\dfrac{xL^{\prime}(x)}{L(x)}\right)\leq\dfrac{3}{2}px^{p-1}L^{p}(x).$
(4.6)

Applying Lemma 2.8, it thus follows from (4.2), (4.4) and (4.6) that

$\mathbb{E}(h(|X|))=\mathbb{E}\left(|X|^{p}L^{p}(|X|)\right)<\infty.$ (4.7)

We have proved that the array $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is
stochastically dominated in the Cesàro sense by a random variable $X$
satisfying (4.7). For $n\geq 1$, set $b_{n}=n^{1/p}\tilde{L}(n^{1/p})$,

$Y_{n,i}=-b_{n}\mathbf{1}(X_{n,i}<-b_{n})+X_{n,i}\mathbf{1}(|X_{n,i}|\leq
b_{n})+b_{n}\mathbf{1}(X_{n,i}>b_{n}),1\leq i\leq n,$

and

$S_{n,k}=\sum_{i=1}^{k}(Y_{n,i}-\mathbb{E}(Y_{n,i})),\ 1\leq k\leq n.$

We will now follow the proof of the implication ((i)$\Rightarrow$(ii)) of
Theorem 3.1 in Anh et al. anh2021marcinkiewicz . Let $\varepsilon>0$ be
arbitrary. For $n\geq 1$,

$\begin{split}&\mathbb{P}\left(\max_{1\leq k\leq
n}\left|\sum_{i=1}^{k}X_{n,i}\right|>\varepsilon
b_{n}\right)\leq\mathbb{P}\left(\max_{1\leq k\leq
n}|X_{n,k}|>b_{n}\right)+\mathbb{P}\left(\max_{1\leq k\leq
n}\left|\sum_{i=1}^{k}Y_{n,i}\right|>\varepsilon b_{n}\right)\\\
&\leq\mathbb{P}\left(\max_{1\leq k\leq
n}|X_{n,k}|>b_{n}\right)+\mathbb{P}\left(\max_{1\leq k\leq
n}|S_{n,k}|>\varepsilon
b_{n}-\sum_{i=1}^{n}\left|\mathbb{E}(Y_{n,i})\right|\right).\end{split}$ (4.8)

Since the array $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is stochastically
dominated in the Cesàro sense by a random variable $X$ satisfying (4.7), we
have from Proposition 2.6 in anh2021marcinkiewicz  that

$\begin{split}\sum_{n=1}^{\infty}n^{-1}\mathbb{P}\left(\max_{1\leq k\leq
n}|X_{n,k}|>b_{n}\right)&\leq\sum_{n=1}^{\infty}n^{-1}\sum_{k=1}^{n}\mathbb{P}\left(|X_{n,k}|>b_{n}\right)\\\
&\leq\sum_{n=1}^{\infty}\mathbb{P}(|X|>b_{n})<\infty.\end{split}$ (4.9)

For $n\geq 1$, it follows from the assumption $\mathbb{E}(X_{n,i})\equiv 0$
and the Cesàro stochastic domination condition that

$\begin{split}\dfrac{\sum_{i=1}^{n}|\mathbb{E}(Y_{n,i})|}{b_{n}}&\leq\dfrac{\sum_{i=1}^{n}\left(\left|\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|\leq
b_{n}))\right|+b_{n}\mathbb{P}(|X_{n,i}|>b_{n})\right)}{b_{n}}\\\
&=\dfrac{\sum_{i=1}^{n}\left(\left|\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|>b_{n}))\right|+b_{n}\mathbb{P}(|X_{n,i}|>b_{n})\right)}{b_{n}}\\\
&\leq\dfrac{2\sum_{i=1}^{n}\mathbb{E}(|X_{n,i}|\mathbf{1}(|X_{n,i}|>b_{n}))}{b_{n}}\\\
&\leq\dfrac{2n\mathbb{E}\left(|X|\mathbf{1}(|X|>b_{n})\right)}{b_{n}}.\end{split}$
(4.10)

For $n$ large enough and for $\omega\in(|X|>b_{n})$, we have (see (3.10) in
anh2021marcinkiewicz )

$\begin{split}\dfrac{n}{b_{n}}&\leq
C|X(\omega)|^{p-1}L^{p}(|X(\omega)|).\end{split}$ (4.11)

Applying (4.10), (4.11), (4.7) and the dominated convergence theorem, we have

$\begin{split}\dfrac{\sum_{i=1}^{n}|\mathbb{E}(Y_{n,i})|}{b_{n}}&\leq
C\mathbb{E}\left(|X|^{\alpha}L^{p}(|X|)\mathbf{1}\left(|X|>b_{n}\right)\right)\to
0\text{ as }n\to\infty.\end{split}$ (4.12)

From (4.8), (4.9) and (4.12), to obtain (4.3), it remains to show that

$\sum_{n=1}^{\infty}n^{-1}\mathbb{P}\left(\max_{1\leq k\leq
n}|S_{n,k}|>b_{n}\varepsilon/2\right)<\infty.$ (4.13)

Applying Markov’s inequality, condition ($H$), and the Cesàro stochastic
domination condition yields

$\begin{split}&\sum_{n=1}^{\infty}\dfrac{1}{n}\mathbb{P}\left(\max_{1\leq
k\leq
n}|S_{n,k}|>b_{n}\varepsilon/2\right)\leq\sum_{n=1}^{\infty}\dfrac{4}{\varepsilon^{2}nb_{n}^{2}}\mathbb{E}\left(\max_{1\leq
k\leq n}|S_{n,k}|\right)^{2}\\\
&\leq\sum_{n=1}^{\infty}\dfrac{C}{nb_{n}^{2}}\sum_{i=1}^{n}\mathbb{E}\left(Y_{n,i}^{2}\right)\\\
&\leq C\sum_{n=1}^{\infty}\dfrac{\left(\mathbb{E}\left(X^{2}\mathbf{1}(|X|\leq
b_{n})\right)+b_{n}^{2}\mathbb{P}(|X|>b_{n})\right)}{b_{n}^{2}}\\\
&=C\sum_{n=1}^{\infty}\left(\mathbb{P}(|X|>b_{n})+\dfrac{\mathbb{E}(X^{2}\mathbf{1}(|X|\leq
b_{n}))}{b_{n}^{2}}\right).\end{split}$ (4.14)

Using the last four lines of (3.13) of anh2021marcinkiewicz  and (3.14) of
anh2021marcinkiewicz , we have

$\begin{split}\sum_{n=1}^{\infty}\left(\mathbb{P}(|X|>b_{n})+\dfrac{\mathbb{E}(X^{2}\mathbf{1}(|X|\leq
b_{n}))}{b_{n}^{2}}\right)\leq C+C\mathbb{E}(|X|^{p}L^{p}(|X|)).\end{split}$
(4.15)

Combining (4.14), (4.15), and (4.7) yields (4.13). ∎

The following corollary establishes rate of convergence in a
Marcinkiewicz–Zygmund-type SLLN for arrays of random variables under a
uniformly bounded moment condition.

###### Corollary 4.2

Let $1\leq p<2$ and let $\nu$ be a fixed positive integer. Let
$\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of mean zero random
variables such that for each $n\geq 1$ fixed, the collection $\\{X_{n,i},1\leq
i\leq n\\}$ satisfies condition $(H)$. If

$\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(|X_{n,i}|^{p}L^{p}(|X_{n,i}|)\log_{\nu}^{(2)}(|X_{n,i}|)\right)<\infty,$
(4.16)

then (4.3) is obtained.

###### Proof

By applying Corollary 2.9, we have from (4.16) that the array
$\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is stochastically dominated in the
Cesàro sense by a random variable $X$ with
$\mathbb{E}(|X|^{p}L^{p}(|X|))<\infty$, that is, (4.2) is satisfied. Applying
Theorem 4.1, we obtain (4.3). ∎

The moment condition (4.16) is almost optimal. The following example shows
that there exists an array of random variables $\\{X_{n,i},1\leq i\leq n,n\geq
1\\}$ such that

$\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(|X_{n,i}|^{p}\log_{\nu}(|X_{n,i}|)\right)<\infty$
(4.17)

for every fixed positive integer $\nu$, but (4.3) fails with
$\tilde{L}(x)\equiv L(x)\equiv 1$.

###### Example 4.3

Let $\nu$ be an arbitrary fixed positive integer and let $1\leq p<2$. Let
$\\{X_{n},n\geq 1\\}$ be a sequence of independent random variables with

$\mathbb{P}(X_{n}=0)=1-\dfrac{1}{n\log_{\nu}(n)},\
\mathbb{P}\left(X_{n}=\pm(n+1)^{1/p}\right)=\dfrac{1}{2n\log_{\nu}(n)},n\geq
1$

and let $X_{n,i}=X_{i},1\leq i\leq n,n\geq 1.$ Then (4.17) is satisfied, and

$\begin{split}\sum_{n=1}^{\infty}\mathbb{P}(|X_{n}|>n^{1/p})&=\sum_{n=1}^{\infty}\dfrac{1}{n\log_{\nu}(n)}=\infty.\end{split}$
(4.18)

If (4.3) (with $\tilde{L}(x)\equiv L(x)\equiv 1$) holds, then

$\sum_{n=1}^{\infty}\dfrac{1}{n}\mathbb{P}\left(\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}X_{i}\right|>\varepsilon n^{1/p}\right)<\infty\text{ for
all }\varepsilon>0.$

This implies that

$\lim_{n\to\infty}\dfrac{\sum_{i=1}^{n}X_{i}}{n^{1/p}}=0\text{ a.s,}$

and thus

$\lim_{n\to\infty}\dfrac{X_{n}}{n^{1/p}}=0\text{ a.s.}$ (4.19)

Applying the Borel–Cantelli lemma, we have from (4.19) that

$\sum_{n=1}^{\infty}\mathbb{P}(|X_{n}|>n^{1/p})<\infty$

contradicting (4.18). Therefore, (4.3) (with $\tilde{L}(x)\equiv L(x)\equiv
1$) must fail.

If we consider sequences of random variables instead of triangular arrays, we
obtain the following Marcinkiewicz–Zygmund-type SLLN with general normalizing
sequences.

###### Corollary 4.4

Let $\\{X_{n},n\geq 1\\}$ be a sequence of mean zero random variables
satisfying condition $(H)$. Let $L(\cdot)$ be a slowly varying function and
let $1\leq p<2$. When $p=1$, we further assume that $L(\cdot)$ is
nondecreasing and $L(x)\geq 1$ for all $x\geq 0$. If

$\int_{0}^{\infty}x^{p-1}L^{p}(x)\left(\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{i}|>x)\right)\operatorname{d}x<\infty,$
(4.20)

then

$\lim_{n\to\infty}\dfrac{\sum_{i=1}^{n}X_{i}}{n^{1/p}\tilde{L}(n^{1/p})}=0\text{
a.s.,}$ (4.21)

where $\tilde{L}(\cdot)$ is the Bruijn conjugate of $L(\cdot)$.

###### Proof

Set

$X_{n,i}=X_{i},1\leq i\leq n,n\geq 1.$

Then (4.20) coincides with (4.2). Applying Theorem 4.1, we have

$\sum_{n=1}^{\infty}\dfrac{1}{n}\mathbb{P}\left(\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}X_{i}\right|>\varepsilon
n^{1/p}\tilde{L}(n^{1/p})\right)<\infty\text{ for all }\varepsilon>0.$ (4.22)

The Marcinkiewicz-Zygmund-type SLLN (4.21) follows from (4.22). ∎

Proof of Theorem 1.2. Since $\\{X_{n},n\geq 1\\}$ is a sequence of AANA random
variables with the sequence of mixing coefficients is in $\ell_{2}$, it
satisfies condition ($H$) (see Lemmas 2.1 and 2.2 of Ko et al. ko2005hajeck ).
Thus Theorem 1.2 follows from Corollary 4.4 by taking $L(x)\equiv 1$. ∎

The following theorem is a significant extension of Theorem 1.3. It
establishes a WLLN for weighted sums from arrays of random variables.

###### Theorem 4.5

Let $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of random variables. Let
$G(\cdot)$ be as in Theorem 4.1 and let $\\{b_{n},n\geq 1\\}$ be a
nondecreasing sequence of positive real numbers satisfying (1.9). Let
$\\{c_{n,i},1\leq i\leq n\\}$ be an array of nonnegative real numbers
satisfying

$0<A_{n}:=\sum_{i=1}^{n}c_{n,i}\leq Cn,\ n\geq 1$ (4.23)

and let

$\hat{G}(x)=\sup_{n\geq 1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\
x\in\mathbb{R},$

where $a_{n,i}=A_{n}^{-1}c_{n,i},\ 1\leq i\leq n,n\geq 1.$ If

$\lim_{k\to\infty}kG(b_{k})=0\ \text{ and }\
\lim_{k\to\infty}k\hat{G}(b_{k})=0,$ (4.24)

then the WLLN

$\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}X_{n,i}\right|\overset{\mathbb{P}}{\to}0\text{ as
}n\to\infty$ (4.25)

is obtained.

###### Remark 4.6

It is clear that in the unweighted case, i.e., $c_{n,i}\equiv 1$, then
$\hat{G}(x)\equiv G(x)$. It is also easy to see that if $\\{X_{n,i},1\leq
i\leq n,n\geq 1\\}$ is stochastically dominated by a random variable $X$ with
$\lim_{k\to\infty}k\mathbb{P}(|X|>b_{k})=0$, then both halves of (4.24) are
fulfilled.

At the first look, the second half of (4.24) may seem to be a technical
condition. However, the following example shows that it cannot be dispensed
with.

###### Example 4.7

Let $0<p<1$, $b_{n}=n^{1/p},\ n\geq 1$ and let $\\{X_{n,i},1\leq i\leq n,n\geq
1\\}$ be an array of random variables such that

$\mathbb{P}(X_{n,i}=-1)=\mathbb{P}(X_{n,i}=1)=1/2\ \text{ for }1\leq i<n,n\geq
2,$

and

$\mathbb{P}(X_{n,n}=-n^{1/p}\log^{-1/p}(n))=\mathbb{P}(X_{n,n}=n^{1/p}\log^{-1/p}(n))=1/2\
\text{ for }n\geq 1.$

Let $\\{c_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of real numbers such
that

$c_{n,i}=0\text{ for }1\leq i<n,n\geq 2,\text{ and }c_{n,n}=n\text{ for }n\geq
1$

and let

$A_{n}=\sum_{i=1}^{n}c_{n,i},\ a_{n,i}=\dfrac{c_{n,i}}{A_{n}},1\leq i\leq
n,n\geq 1.$

Then (4.23) is satisfied since $A_{n}\equiv n$. Let

$G(x)=\sup_{n\geq 1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x),\
x\in\mathbb{R},$

and

$\hat{G}(x)=\sup_{n\geq 1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\
x\in\mathbb{R}.$

For $n\geq 1$, we have

$\begin{split}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(|X_{n,i}|^{p}\log(|X_{n,i}|)\right)&\leq
1+\dfrac{1}{n}\mathbb{E}(|X_{n,n}|^{p}\log(|X_{n,n}|))\\\
&=1+\dfrac{1}{\log(n)}\left(\dfrac{\log(n)-\log(\log(n))}{p}\right)\\\ &\leq
1+\dfrac{1}{p}<\infty.\end{split}$

Therefore, $\\{|X_{n,i}|^{p},1\leq i\leq n,n\geq 1\\}$ is uniformly integrable
in the Cesàro sense by the de La Vallée Poussin criterion for the Cesàro
uniform integrability. Then by Corollary 3.2, $\\{X_{n,i},1\leq i\leq n,n\geq
1\\}$ is stochastically dominated in the Cesàro sense by a random variable $X$
with distribution function

$F(x)=1-\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x)=1-G(x),\ x\in\mathbb{R},$

and the first half of (4.24) (with $b_{n}\equiv n^{1/p}$) is satisfied.
However, the second half of (4.24) (with $b_{n}\equiv n^{1/p}$) fails since

$\hat{G}(x)=\sup_{n\geq
1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x)=\sup_{n\geq
1}\mathbb{P}(|X_{n,n}|>x)=1\text{ for all }x\in\mathbb{R}.$

For $n\geq 1$, we have with probability $1$,

$\begin{split}\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}X_{n,i}\right|&=\dfrac{1}{n^{1/p}}c_{n,n}|X_{n,n}|\\\
&=\dfrac{n}{\log^{1/p}(n)}\to\infty\end{split}$

therefore the WLLN (4.25) also fails.

Proof of Theorem 4.5. From (1.9), we have $b_{n}\to\infty$ (see (4.31) below).
Since $G(x)$ and $\hat{G}(x)$ are nonincreasing, it follows from (4.24) that
$\lim_{x\to\infty}G(x)=0$ and $\lim_{x\to\infty}\hat{G}(x)=0.$ By Corollary
2.4, $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ is stochastically dominated in the
Cesàro sense by a random variable $X$, and by Theorem 2.1, $\\{X_{n,i},1\leq
i\leq n,n\geq 1\\}$ is $\\{a_{n,i}\\}$-stochastically dominated by a random
variable $Y$. The distribution functions of $X$ and $Y$, respectively, are

$F_{X}(x)=1-G(x)\ \text{ and }\ F_{Y}(x)=1-\hat{G}(x),\ x\in\mathbb{R}.$

Thus

$\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>x)=\mathbb{P}(|X|>x),\
x\in\mathbb{R},$ (4.26)

and

$\sup_{n\geq
1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x)=\mathbb{P}(|Y|>x),\
x\in\mathbb{R},$ (4.27)

and so (4.24) becomes

$\lim_{k\to\infty}k\mathbb{P}(|X|>b_{k})=0\ \text{ and }\
\lim_{k\to\infty}k\mathbb{P}(|Y|>b_{k})=0.$ (4.28)

For $n\geq 1$, set

$Y_{n,i}=X_{n,i}\mathbf{1}(|X_{n,i}|\leq b_{n}),\ 1\leq i\leq n.$

We first verify that

$\dfrac{\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}\left(X_{n,i}-Y_{n,i}\right)\right|}{b_{n}}\overset{\mathbb{P}}{\to}0\text{
as }n\rightarrow\infty.$ (4.29)

To see this, let $\varepsilon>0$ be arbitrary. Then we have from (4.26) and
the first half of (4.28) that

$\begin{split}\mathbb{P}\left(\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}\left(X_{n,i}-Y_{n,i}\right)\right|>b_{n}\varepsilon\right)&\leq\mathbb{P}\left(\bigcup_{i=1}^{n}(X_{n,i}\not=Y_{n,i})\right)\\\
&\leq\sum_{i=1}^{n}\mathbb{P}(|X_{n,i}|>b_{n})\\\ &\leq
n\mathbb{P}(|X|>b_{n})\to 0\text{ as }n\to\infty\end{split}$

thereby proving $\eqref{wl133}$.

Next, it will be shown that

$\dfrac{\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}Y_{n,i}\right|}{b_{n}}\overset{\mathbb{P}}{\to}0\text{
as }n\rightarrow\infty.$ (4.30)

To accomplish this, we first recall that (1.9) implies (see Remark 2.4 (i) in
Boukhari boukhari2021weak )

$\dfrac{n}{b_{n}}\to 0\text{ as }n\to\infty.$ (4.31)

Set $b_{0}=0$. Again, let $\varepsilon>0$ be arbitrary. Then

$\begin{split}\mathbb{P}\left(\max_{1\leq j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}Y_{n,i}\right|>b_{n}\varepsilon\right)&\leq\dfrac{1}{b_{n}\varepsilon}\mathbb{E}\left(\max_{1\leq
j\leq n}\left|\sum_{i=1}^{j}c_{n,i}Y_{n,i}\right|\right)\\\
&\leq\dfrac{1}{b_{n}\varepsilon}\sum_{i=1}^{n}c_{n,i}\mathbb{E}(|Y_{n,i}|)\\\
&\leq\dfrac{1}{b_{n}\varepsilon}\sum_{i=1}^{n}c_{n,i}\int_{0}^{b_{n}}\mathbb{P}(|X_{n,i}|>x)\operatorname{d}x\\\
&=\dfrac{A_{n}}{b_{n}\varepsilon}\sum_{i=1}^{n}a_{n,i}\int_{0}^{b_{n}}\mathbb{P}(|X_{n,i}|>x)\operatorname{d}x\\\
&\leq\dfrac{A_{n}}{b_{n}\varepsilon}\int_{0}^{b_{n}}\mathbb{P}(|Y|>x)\operatorname{d}x\\\
&=\dfrac{A_{n}}{b_{n}\varepsilon}\sum_{k=1}^{n}\int_{b_{k-1}}^{b_{k}}\mathbb{P}(|Y|>x)\operatorname{d}x\\\
&\leq\dfrac{Cn}{b_{n}\varepsilon}\sum_{k=1}^{n}\dfrac{b_{k}-b_{k-1}}{k}k\mathbb{P}(|Y|>b_{k-1}),\end{split}$
(4.32)

where we have applied Markov’s inequality in the first inequality, (4.27) in
the fourth inequality, and (4.23) in the last inequality. Now when $n\geq 2$,

$\begin{split}\dfrac{n}{b_{n}}\sum_{k=1}^{n}\dfrac{b_{k}-b_{k-1}}{k}&=\dfrac{n}{b_{n}}\left(\sum_{k=1}^{n-1}\dfrac{b_{k}}{k(k+1)}+\dfrac{b_{n}}{n}\right)\\\
&\leq\dfrac{n}{b_{n}}\left(\sum_{k=1}^{n-1}\dfrac{b_{k}}{k^{2}}+\dfrac{b_{n}}{n}\right)\\\
&\leq C\ \text{ (by \ref{wl01})},\end{split}$

for all fixed $k$,

$\dfrac{n}{b_{n}}\left(\dfrac{b_{k}-b_{k-1}}{k}\right)\to 0\text{ as
}n\to\infty\ \text{ (by \eqref{wl137})},$

and for $k\geq 2$,

$k\mathbb{P}(|Y|>b_{k-1})\leq 2(k-1)\mathbb{P}(|Y|>b_{k-1})\to 0\text{ as
}k\to\infty\ \text{ (by the second half of \ref{wl127})}.$

Thus by the Toeplitz lemma

$\dfrac{n}{b_{n}}\sum_{k=1}^{n}\dfrac{b_{k}-b_{k-1}}{k}k\mathbb{P}(|Y|>b_{k-1})\to
0\text{ as }n\to\infty$

and (4.30) then follows from (4.32). Combining (4.29) and (4.30) yields
(4.25).∎

Proof of Theorem 1.3. Set

$c_{n,i}=1,\ X_{n,i}=X_{i},1\leq i\leq n,n\geq 1.$

Then (1.10) coincides with (4.24). Theorem 1.3 follows from Theorem 4.5.∎

We will now present an example to illustrate Theorem 1.3. This example shows
that for $0<p<1$ and $b_{n}=n^{1/p}$, there exists a sequence of random
variables $\\{X_{n},n\geq 1\\}$ with no stochastically dominating random
variable and condition (1.10) is satisfied. In this example, we also show that
(1.3) (with $0<p<1$) holds but (1.2) (with $0<p<1$) does not.

###### Example 4.8

Let $0<p<1$, $b_{n}=n^{1/p},\ n\geq 1$ and let $\\{X_{n},n\geq 1\\}$ be a
sequence of random variables such that

$\mathbb{P}(X_{n}=-1)=\mathbb{P}(X_{n}=1)=1/2\ \text{ for }n\not=2^{m},\ m\geq
0,$ (4.33)

and

$\mathbb{P}\left(X_{2^{m}}=-2^{m/p}/m^{1/p}\right)=\mathbb{P}\left(X_{2^{m}}=2^{m/p}/m^{1/p}\right)=1/2\text{
for }m\geq 0.$ (4.34)

Then

$\sup_{n\geq 1}\mathbb{P}(|X_{n}|>x)=\sup_{m\geq
1}\mathbb{P}(|X_{2^{m}}|>x)=1\text{ for all }x\geq 1.$

Thus there is no random variable $X$ such that the sequence $\\{X_{n},n\geq
1\\}$ is stochastically dominated by $X$, and so we cannot apply Theorem 2.1
of Boukhari boukhari2021weak .

Now, for $n\geq 1$, let $m\geq 0$ be such that $2^{m}\leq n<2^{m+1}$. Then

$\begin{split}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(|X_{i}|^{p}\log(|X_{i}|)\right)&\leq
1+\dfrac{1}{2^{m}}\sum_{i=0}^{m}\mathbb{E}\left(|X_{2^{i}}|^{p}\log(|X_{2^{i}}|)\right)\\\
&=1+\dfrac{1}{2^{m}}\sum_{i=0}^{m}\dfrac{2^{i}}{i}\left(\dfrac{i-\log(i)}{p}\right)\\\
&\leq 1+\dfrac{2}{p}<\infty.\end{split}$ (4.35)

Therefore, $\\{|X_{n}|^{p},n\geq 1\\}$ is uniformly integrable in the Cesàro
sense by the de La Vallée Poussin criterion for the Cesàro uniform
integrability. Then by Corollary 3.2, the sequence $\\{X_{n},n\geq 1\\}$ is
stochastically dominated in the Cesàro sense by a random variable $X$ with
distribution function

$F(x)=1-\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{i}|>x):=1-G(x),\ x\in\mathbb{R},$
(4.36)

and condition (1.10) (with $b_{n}\equiv n^{1/p}$) is satisfied. It thus
follows from Theorem 1.3 that the WLLN (1.11) holds.

Finally, it is clear that $\mathbb{P}(|X_{n}|^{p}>n)=0$ for all $n\geq 1$ so
that (1.3) holds. We will show that (1.2) (with $0<p<1$) is not satisfied. To
see this, for $x\geq 1$, let $n_{x}$ be the smallest integer such that

$\dfrac{2^{n_{x}}}{n_{x}}>x^{p}.$

Then for $x\geq 1$,

$\dfrac{2^{n_{x}}}{n_{x}}>x^{p}\geq\dfrac{2^{n_{x}-1}}{n_{x}-1}$ (4.37)

and it follows from (4.37) that there exists $\varepsilon_{0}>0$ such that

$\dfrac{1}{2^{n_{x}}}>\dfrac{\varepsilon_{0}}{x^{p}\log(x)}\ \text{ for all
large }x.$ (4.38)

Combining (4.33)–(4.36) and (4.38), we have

$G(x)=\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{P}(|X_{i}|>x)\geq\dfrac{1}{2^{n_{x}}}\sum_{i=1}^{2^{n_{x}}}\mathbb{P}(|X_{i}|>x)=\dfrac{1}{2^{n_{x}}}\geq\dfrac{\varepsilon_{0}}{x^{p}\log(x)}\text{
for all large }x$ (4.39)

and it follows from (4.39) that (1.2) (with $0<p<1$) fails. Thus we cannot
apply Remark 3 of Chandra and Ghosal chandra1996extensions  to obtain the
Marcinkiewicz–Zygmund SLLN for the case $0<p<1$.

Now, we discuss about the normalizing sequences in the WLLN in Theorem 1.3. We
note that for $0<p<2$ and $b_{n}\equiv n^{1/p}$, (1.9) is fulfilled if $0<p<1$
but it fails to hold if $1\leq p<2$. For the case where $1\leq p<2$, we also
have to require some dependence structures to obtain the WLLN (see Boukhari
boukhari2021weak  for a counterexample). Kruglov kruglov2011generalization
established a Kolmogorov–Feller-type WLLN for sequences of negatively
associated identically distributed random variables with normalizing sequences
$b_{n},n\geq 1$ satisfying

$\sum_{i=1}^{n}\dfrac{b_{i}^{2}}{i^{2}}=O\left(\dfrac{b_{n}^{2}}{n}\right).$
(4.40)

It was observed by Kruglov kruglov2011generalization  that (4.40) holds for
the case $b_{n}\equiv n^{1/p}L(n)$, where $1\leq p<2$ and $L(\cdot)$ is a
slowly varying function. The following theorem appears to be new even when the
underlying random variables are independent. It also extends the sufficient
part of Theorem 1 of Kruglov kruglov2011generalization . The proof of Theorem
4.9 can be obtained by proceeding in a similar manner as that of Theorem 4.5:
We firstly use the nondecreasing truncation as in the proof of Theorem 4.1,
and use the maximal inequality (4.1) instead of the triangular inequality (in
the place of the second inequality in (4.32)), and then change the other
places accordingly. We leave the details to the interested reader.

###### Theorem 4.9

Let $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of random variables such
that for each $n\geq 1$ fixed, the collection $\\{X_{n,i},1\leq i\leq n\\}$
satisfies condition $(H)$ and let $G(\cdot)$ be as in Theorem 4.1. Let
$\\{b_{n},n\geq 1\\}$ be a nondecreasing sequence of positive real numbers
satisfying (4.40) and let $\\{c_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of
nonnegative real numbers satisfying

$0<A_{n}:=\sum_{i=1}^{n}c_{n,i}^{2}\leq Cn,\ n\geq 1.$ (4.41)

Let

$\hat{G}(x)=\sup_{n\geq 1}\sum_{i=1}^{n}a_{n,i}\mathbb{P}(|X_{n,i}|>x),\
x\in\mathbb{R},$

where $a_{n,i}=A_{n}^{-1}c_{n,i}^{2},\ 1\leq i\leq n,n\geq 1.$ If

$\lim_{k\to\infty}kG(b_{k})=0\ \text{ and }\
\lim_{k\to\infty}k\hat{G}(b_{k})=0,$ (4.42)

then the WLLN

$\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}c_{n,i}\left(X_{n,i}-\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|\leq
b_{n}))\right)\right|\overset{\mathbb{P}}{\to}0\text{ as }n\to\infty$

is obtained.

We now apply Corollary 3.2 and Theorem 4.9 to obtain a WLLN for arrays of
random variables under the Cesàro uniform integrability condition. For
simplicity and since it is just meant to be an illustration, we only consider
the unweighted case, i.e., the case where $c_{n,i}\equiv 1$.

###### Corollary 4.10

Let $1\leq p<2$ and let $\\{X_{n,i},1\leq i\leq n,n\geq 1\\}$ be an array of
random variables such that for each $n\geq 1$ fixed, the collection
$\\{X_{n,i},1\leq i\leq n\\}$ satisfies condition $(H)$. Let $G(\cdot)$ be as
in Theorem 4.1 and $L(\cdot)$ be a slowly varying function. Let
$\tilde{L}(\cdot)$ be the Bruijin conjugate of $L(\cdot)$. In the case $p=1$,
we further assume that $L(x)$ is nondecreasing and $L(x)\geq 1$ for all $x\geq
0$. If $\\{|X_{n,i}|^{p}L(|X_{n,i}|^{p}),1\leq i\leq n,n\geq 1\\}$ is
uniformly integrable in the Cesàro sense, that is,

$\lim_{a\to\infty}\sup_{n\geq
1}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left(|X_{n,i}|^{p}L(|X_{n,i}|^{p})\mathbf{1}(|X_{n,i}|>a)\right)=0,$
(4.43)

then the WLLN

$\dfrac{1}{n^{1/p}\tilde{L}^{1/p}(n)}\max_{j\leq
n}\left|\sum_{i=1}^{j}\left(X_{n,i}-\mathbb{E}(X_{n,i})\right)\right|\overset{\mathbb{P}}{\to}0\text{
as }n\to\infty$ (4.44)

is obtained.

###### Proof

In Theorem 4.9, if we choose $c_{n,i}\equiv 1$, then (4.41) is automatic, and
$\hat{G}(x)\equiv G(x)$. By Corollary 3.2, it follows from (4.43) that (4.42)
holds with $b_{n}\equiv n^{1/p}\tilde{L}^{1/p}(n)$. Applying Theorem 4.9, we
obtain

$\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}\left(X_{n,i}-\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|\leq
b_{n}))\right)\right|\overset{\mathbb{P}}{\to}0\text{ as }n\to\infty.$

To obtain (4.44), it remains to show that

$\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|>b_{n}))\right|\to
0\text{ as }n\to\infty.$ (4.45)

Since the function $x^{p-1}L(x)$ is nondecreasing, we have

$\begin{split}&\dfrac{1}{b_{n}}\max_{j\leq
n}\left|\sum_{i=1}^{j}\mathbb{E}(X_{n,i}\mathbf{1}(|X_{n,i}|>b_{n}))\right|\leq\dfrac{1}{b_{n}}\sum_{i=1}^{n}\mathbb{E}(|X_{n,i}|\mathbf{1}(|X_{n,i}|>b_{n}))\\\
&\leq\dfrac{1}{b_{n}}\sum_{i=1}^{n}\dfrac{\mathbb{E}(|X_{n,i}|^{p}L(|X_{n,i}|^{p})\mathbf{1}(|X_{n,i}|>b_{n}))}{b_{n}^{p-1}L(b_{n}^{p})}\\\
&=\left(\dfrac{1}{\tilde{L}(n)L\left(n\tilde{L}(n)\right)}\right)\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}(|X_{n,i}|^{p}L(|X_{n,i}|^{p})\mathbf{1}(|X_{n,i}|>b_{n})).\end{split}$
(4.46)

Now, from the second half of (2.5) we have
$\lim_{n\to\infty}\tilde{L}(n)L\left(n\tilde{L}(n)\right)=1$, and from (4.43)
we have

$\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\mathbb{E}(|X_{n,i}|^{p}L(|X_{n,i}|^{p})\mathbf{1}(|X_{n,i}|>b_{n}))=0.$

Therefore (4.45) follows from (4.46). The proof of the corollary is completed.
∎

##  5 Conclusions and open problems

In Section 4, our results on the concept of $\\{a_{n,i}\\}$-stochastic
domination are applied to obtain the WLLNs for weighted sums. The results on
the Cesàro stochastic domination case are applied to obtain rate of
convergence in the SLLN with general normalizing sequences under the
Chandra–Ghosal-type condition, and these results help us to remove an
assumption of a SLLN established by Chandra and Ghosal chandra1996extensions .
The results on the concept of $\\{a_{n,i}\\}$-stochastic domination may also
be useful in proving weighted SLLNs of Chandra and Ghosal chandra1996strong
as we will describe as follows.

Let $1\leq p<2$ and let $\\{a_{n},n\geq 1\\}$ be a sequence of positive real
numbers with

$A_{n}:=\sum_{i=1}^{n}a_{i}\to\infty\ \text{ as }n\to\infty.$

Let $\\{X_{n},n\geq 1\\}$ be a sequence of mean zero random variables which
satisfies suitable dependence conditions. Let $G(x)$ be as in Theorem 1.1 and
let

$\tilde{G}(x)=\sup_{n\geq
1}\left(\sum_{i=1}^{n}a_{i}^{1/p}\right)^{-1}\sum_{i=1}^{n}a_{i}^{1/p}\mathbb{P}(|X_{i}|>x),x\in\mathbb{R}.$

Chandra and Ghosal chandra1996strong  considered the following three
conditions (see (2.14)–(2.16) in chandra1996strong ):

$\int_{0}^{\infty}x^{p-1}G(x)\operatorname{d}x<\infty,$ (5.1)
$\int_{0}^{\infty}x^{p-1}\tilde{G}(x)\operatorname{d}x<\infty,$ (5.2)

and

$\sum_{n=1}^{\infty}\mathbb{P}(|X_{n}|^{p}>A_{n}/a_{n})<\infty.$ (5.3)

Chandra and Ghosal (see Theorems 2.6 and 2.7 in chandra1996strong ) proved
that if (5.1), (5.2) and (5.3) are all satisfied, then the weighted
Marcinkiewicz–Zygmund SLLN

$\dfrac{\sum_{i=1}^{n}a_{i}^{1/p}X_{i}}{A_{n}^{1/p}}\to 0\text{ a.s. as
}n\to\infty$

is obtained. In view of Theorems 1.2 and 4.5, we state an open problem as to
whether or not the Chandra and Ghosal result mentioned above still holds
without Condition (5.3).

For $n\geq 1$, let

$a_{n,i}=\left(\sum_{i=1}^{n}a_{i}^{1/p}\right)^{-1}a_{i}^{1/p},\text{ }1\leq
i\leq n.$

Since $\tilde{G}(x)$ is nonincreasing, it follows from (5.2) that
$\lim_{x\to\infty}\tilde{G}(x)=0$. By Theorem 2.1, $\\{X_{n},n\geq 1\\}$ is
$\\{a_{n,i}\\}$-stochastically dominated by a random variable $X$ with
distribution function $F(x)=1-\tilde{G}(x)$, and (5.2) becomes
$\mathbb{E}(|X|^{p})<\infty$. In view of the proof of Theorem 4.5, the results
on the concept of $\\{a_{n,i}\\}$-stochastic domination established in
Sections 2 and 3 may help in answering the above open problem.

Finally, we present an open problem concerning Corollary 4.10. For the case
where $L(x)\equiv\tilde{L}(x)\equiv 1$, we can obtain convergence in mean of
order $p$ in (4.44) (see, e.g., Theorem 1 in chandra1989uniform , Theorem 4 in
ordonezcabrera1994convergence , Theorem 2.1 in thanh2005lp ). However, the
methods in chandra1989uniform ; ordonezcabrera1994convergence ; thanh2005lp
do not seem to work for general slowly varying function $L(\cdot)$, even with
assumption that the underlying random variables are independent. It is an open
problem as to whether or not convergence in mean of order $p$ prevails in the
conclusion (4.44).

Conflict of interest: The author has no conflict of interest.

Funding: The author did not receive support from any organization for this
work.

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