On the convergence of series of moments for row sums of random variables

TL;DR

This paper investigates the convergence of series involving moments of row sums of random variables in a triangular array, extending results to various dependence structures.

Contribution

It establishes conditions for the convergence of series of moments of centered row sums under general dependence assumptions.

Findings

Series convergence under various dependence structures.

Conditions involving moments and sequences for convergence.

General setting broadening previous results.

Abstract

Given a triangular array $\left\{X_{n,k}, \, 1 \leqslant k \leqslant n, n \geqslant 1 \right\}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k} \rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences $\{b_{n} \}$, $\{c_{n} \}$ of positive real numbers, we shall prove that $\sum_{n=1}^\infty c_n \mathbb{E} \left[ |\sum_{k=1}^n (X_{n,k} - \mathbb{E} \, X_{n,k})| / b_n - \varepsilon \right]_+^p < \infty$, where $x_+ = \max(x,0)$. Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.

Equations

$$\left\{\begin{array}[]{l}\mathbb{E}\lvert X_{1}\rvert^{p}<\infty\;\;\text{if $p>qr$}\\[5.0pt] \mathbb{E}\lvert X_{1}\rvert^{qr}\log(1+\lvert X_{1}\rvert)<\infty\;\;\text{if $p=qr$}\\[5.0pt] \mathbb{E}\lvert X_{1}\rvert^{qr}<\infty\;\;\text{if $p<qr$}\end{array}\right.$$

$${\int }_{\epsilon }^{\infty }\sum _{n=1}^{\infty }{n}^{r-2}\mathbb{P}\left\{\left|\sum _{k=1}^n{X}_k\right|>x^{1/p}{n}^{1/q}\right\}\mathrm{d}x<\infty \text{for all ε>0}.$$

$${\int }_{\epsilon }^{\infty }\sum _{n=1}^{\infty }{c}_n\mathbb{P}\left\{\left|X_n\right|>x^{1/p}{b}_n\right\}\mathrm{d}x<\infty \text{for all ε>0}$$

$$\sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\max \left(\frac{|X_n|}{b_n}-\epsilon ,0\right)\right]}^p<\infty \text{for all ε>0}$$

$$\sum _{n=1}^{\infty }{n}^{r-2}\mathbb{E}{\left(n^{-1/q}\left|\sum _{k=1}^n{X}_k\right|-\epsilon \right)}_{+}^p<\infty \text{for all ε>0}.$$

$$\sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}^p<\infty \text{for all ε>0}$$

$$\mathbb{E}{\left|\sum _{k=1}^n\left[g_t(X_{n,k})-\mathbb{E}{g}_t(X_{n,k})\right]\right|}^q⩽{\alpha }_n\sum _{k=1}^n\mathbb{E}|g_t(X_{n,k}){|}^q$$

$$\sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}^p<\infty$$

$$\begin{array}{rl}& \mathbb{E}{\left[\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|-\epsilon b_n\right]}_{+}^p\\ & ={\int }_0^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>\epsilon b_n+t^{1/p}\right\}\mathrm{d}t\\ & ⩽b_n^p\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>\epsilon b_n\right\}+{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>t^{1/p}\right\}\mathrm{d}t.\end{array}$$

$$\begin{array}{rl}& \mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>\epsilon b_n\right\}\\ & ⩽\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}^{\prime }-\mathbb{E}{X}_{n,k}^{\prime })\right|>\frac{\epsilon b_n}{2}\right\}+\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}^{\prime \prime }-\mathbb{E}{X}_{n,k}^{\prime \prime })\right|>\frac{\epsilon b_n}{2}\right\}\\ & ⩽\frac{2^q}{{\epsilon }^q{b}_n^q}\mathbb{E}{\left|\sum _{k=1}^n(X_{n,k}^{\prime }-\mathbb{E}{X}_{n,k}^{\prime })\right|}^q+\frac{2}{\epsilon b_n}\mathbb{E}\left|\sum _{k=1}^n(X_{n,k}^{\prime \prime }-\mathbb{E}{X}_{n,k}^{\prime \prime })\right|\\ & ⩽\frac{2^q{\alpha }_n}{{\epsilon }^q{b}_n^q}\sum _{k=1}^n\mathbb{E}|X_{n,k}^{\prime }{|}^q+\frac{4}{\epsilon b_n}\sum _{k=1}^n\mathbb{E}|X_{n,k}^{\prime \prime }|\\ & ⩽\frac{2^{2q-1}{\alpha }_n}{{\epsilon }^q{b}_n^q}\sum _{k=1}^n\left[\mathbb{E}|X_{n,k}{|}^q{I}_{\left\{|X_{n,k}|⩽b_n\right\}}+b_n^q\mathbb{P}\left\{|X_{n,k}|>b_n\right\}\right]+\frac{4}{\epsilon b_n}\sum _{k=1}^n\mathbb{E}|X_{n,k}|I_{\left\{|X_{n,k}|>b_n\right\}}\\ & =\frac{2^{2q-1}{\alpha }_n}{{\epsilon }^q{b}_n^q}\sum _{k=1}^n{\int }_0^{b_n^q}\mathbb{P}\left\{|X_{n,k}{|}^q>t\right\}\mathrm{d}t+\frac{4}{\epsilon b_n}\sum _{k=1}^n\mathbb{E}|X_{n,k}|I_{\left\{|X_{n,k}|>b_n\right\}}.\end{array}$$

$${\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>t^{1/p}\right\}\mathrm{d}t$$

$$⩽{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(Y_{n,k}^{\prime }-\mathbb{E}{Y}_{n,k}^{\prime })\right|>\frac{t^{1/p}}{2}\right\}\mathrm{d}t+{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(Y_{n,k}^{\prime \prime }-\mathbb{E}{Y}_{n,k}^{\prime \prime })\right|>\frac{t^{1/p}}{2}\right\}\mathrm{d}t.$$

$${\int }_{b_n^p}^{\infty }$$

$$⩽2^q{\int }_{b_n^p}^{\infty }{t}^{-q/p}\mathbb{E}{\left|\sum _{k=1}^n\left(Y_{n,k}^{\prime }-\mathbb{E}{Y}_{n,k}^{\prime }\right)\right|}^q\mathrm{d}t$$

$$⩽2^q{\alpha }_n{\int }_{b_n^p}^{\infty }{t}^{-q/p}\sum _{k=1}^n\mathbb{E}|Y_{n,k}^{\prime }{|}^q\mathrm{d}t$$

$$⩽2^{2q-1}{\alpha }_n{\int }_{b_n^p}^{\infty }{t}^{-q/p}\sum _{k=1}^n\left[\mathbb{E}|X_{n,k}{|}^q{I}_{\left\{|X_{n,k}|⩽t^{1/p}\right\}}+t^{q/p}\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\right]\mathrm{d}t$$

$$=2^{2q-1}q{\alpha }_n\sum _{k=1}^n{\int }_{b_n^p}^{\infty }{t}^{-q/p}{\int }_0^{t^{1/p}}{s}^{q-1}\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}\mathrm{d}s\mathrm{d}t$$

$$=2^{2q-1}q{\alpha }_n\sum _{k=1}^n{\int }_0^{\infty }{s}^{q-1}\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}{\int }_{\max \left(b_n^p,s^p\right)}^{\infty }{t}^{-q/p}\mathrm{d}t\mathrm{d}s$$

$$=2^{2q-1}q{\alpha }_n\sum _{k=1}^n\left[\frac{p{b}_n^{p-q}}{q-p}{\int }_0^{b_n}{s}^{q-1}\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}\mathrm{d}s+\frac{p}{q-p}{\int }_{b_n}^{\infty }{s}^{p-1}\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}\mathrm{d}s\right]$$

$$=\frac{p{2}^{2q-1}{\alpha }_n{b}_n^{p-q}}{q-p}\sum _{k=1}^n{\int }_0^{b_n^q}\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/q}\right\}\mathrm{d}t+\frac{q{2}^{2q-1}{\alpha }_n}{q-p}\sum _{k=1}^n{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\mathrm{d}t.$$

$$\begin{array}{rl}{\int }_{b_n^p}^{\infty }\mathbb{P}& \left\{\left|\sum _{k=1}^n\left(Y_{n,k}^{\prime \prime }-\mathbb{E}{Y}_{n,k}^{\prime \prime }\right)\right|>\frac{t^{1/p}}{2}\right\}\mathrm{d}t\\ & ⩽4{\int }_{b_n^p}^{\infty }{t}^{-1/p}\sum _{k=1}^n\mathbb{E}\left|Y_{n,k}^{\prime \prime }\right|\mathrm{d}t\\ & ⩽4\sum _{k=1}^n{\int }_{b_n^p}^{\infty }{t}^{-1/p}\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>t^{1/p}\right\}}\mathrm{d}t\\ & =4\sum _{k=1}^n\left({\int }_{b_n^p}^{\infty }{t}^{-1/p}{\int }_{t^{1/p}}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}\mathrm{d}s\mathrm{d}t+{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\mathrm{d}t\right)\\ & =4\sum _{k=1}^n\left({\int }_{b_n}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}{\int }_{b_n^p}^{s^p}{t}^{-1/p}\mathrm{d}t\mathrm{d}s+{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\mathrm{d}t\right)\\ & ⩽4\sum _{k=1}^n\left(\frac{p}{p-1}{\int }_{b_n}^{\infty }{s}^{p-1}\mathbb{P}\left\{\left|X_{n,k}\right|>s\right\}\mathrm{d}s+{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\mathrm{d}t\right)\\ & =\frac{4p}{p-1}\sum _{k=1}^n{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t^{1/p}\right\}\mathrm{d}t.\end{array}$$

$$\begin{array}{rl}& \sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}^p\\ & =\sum _{n=1}^{\infty }\frac{c_n}{b_n^p}\mathbb{E}{\left[\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|-\epsilon b_n\right]}_{+}^p\\ & ⩽\sum _{n=1}^{\infty }\left(c_n\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>\epsilon b_n\right\}+\frac{c_n}{b_n^p}{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>t^{1/p}\right\}\mathrm{d}t\right)\\ & ⩽\left(\frac{2^{2q-1}}{{\epsilon }^q}+\frac{p{2}^{2q-1}}{q-p}\right)\sum _{n=1}^{\infty }\sum _{k=1}^n\frac{{\alpha }_n{c}_n}{b_n^q}{\int }_0^{b_n^q}\mathbb{P}\left\{{\left|X_{n,k}\right|}^q>t\right\}\mathrm{d}t+\frac{4}{\epsilon }\sum _{n=1}^{\infty }\sum _{k=1}^n\frac{c_n}{b_n}\mathbb{E}|X_{n,k}|I_{\left\{|X_{n,k}|>b_n\right\}}\\ & +\left(\frac{q{2}^{2q-1}}{q-p}+\frac{4p}{p-1}\right)\sum _{n=1}^{\infty }\sum _{k=1}^n\frac{(1+{\alpha }_n)c_n}{b_n^p}{\int }_{b_n^p}^{\infty }\mathbb{P}\left\{{\left|X_{n,k}\right|}^p>t\right\}\mathrm{d}t\\ & <\infty ,\end{array}$$

$$\sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}<\infty$$

$$\left|\sum _{k=1}^n\left(Y_{n,k}^{\prime \prime }-\mathbb{E}{Y}_{n,k}^{\prime \prime }\right)\right|$$

$$⩽\sum _{k=1}^n\left(\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>t\right\}}+\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>t\right\}}\right)$$

$$⩽\sum _{k=1}^n\left(\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}+\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}\right).$$

$$\begin{array}{rl}{\int }_{b_n}^{\infty }\mathbb{P}& \left\{\left|\sum _{k=1}^n\left(Y_{n,k}^{\prime \prime }-\mathbb{E}{Y}_{n,k}^{\prime \prime }\right)\right|>\frac{t}{2}\right\}\mathrm{d}t\\ & ⩽{\int }_{b_n}^{\infty }\mathbb{P}\left\{\sum _{k=1}^n\left(\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}+\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}\right)>\frac{t}{2}\right\}\mathrm{d}t\\ & =2{\int }_{\frac{b_n}{2}}^{\infty }\mathbb{P}\left\{\sum _{k=1}^n\left(\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}+\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}\right)>s\right\}\mathrm{d}s\\ & ⩽2\mathbb{E}\left[\sum _{k=1}^n\left(\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}+\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}\right)\right]\\ & =4\sum _{k=1}^n\mathbb{E}\left|X_{n,k}\right|I_{\left\{|X_{n,k}|>b_n\right\}}\end{array}$$

$$c_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}$$

$$=\frac{c_n}{b_n}\mathbb{E}{\left[\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|-\epsilon b_n\right]}_{+}$$

$$⩽\left(\frac{2^{2q-1}}{{\epsilon }^q}+\frac{2^{2q-1}}{q-1}\right)\sum _{k=1}^n\frac{{\alpha }_n{c}_n}{b_n^q}{\int }_0^{b_n^q}\mathbb{P}\left\{{\left|X_{n,k}\right|}^q>t\right\}\mathrm{d}t+\left(\frac{4}{\epsilon }+4\right)\sum _{k=1}^n\frac{c_n}{b_n}\mathbb{E}|X_{n,k}|I_{\left\{|X_{n,k}|>b_n\right\}}$$

$$+\frac{q{2}^{2q-1}}{q-1}\sum _{k=1}^n\frac{{\alpha }_n{c}_n}{b_n}{\int }_{b_n}^{\infty }\mathbb{P}\left\{\left|X_{n,k}\right|>t\right\}\mathrm{d}t$$

$$\mathbb{E}{\left|\sum _{k=1}^n\left[g_t(X_{n,k})-\mathbb{E}{g}_t(X_{n,k})\right]\right|}^q⩽{\beta }_n\sum _{k=1}^n\mathbb{E}|g_t(X_{n,k}){|}^q+{\xi }_n{\left[\sum _{k=1}^n\mathbb{E}|g_t(X_{n,k}){|}^2\right]}^{q/2}$$

$$\sum _{n=1}^{\infty }{c}_n\mathbb{E}{\left[\frac{\left|{\sum }_{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|}{b_n}-\epsilon \right]}_{+}^p<\infty$$

$$\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}-\mathbb{E}{X}_{n,k})\right|>\epsilon b_n\right\}$$

$$⩽\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}^{\prime }-\mathbb{E}{X}_{n,k}^{\prime })\right|>\frac{\epsilon b_n}{2}\right\}+\mathbb{P}\left\{\left|\sum _{k=1}^n(X_{n,k}^{\prime \prime }-\mathbb{E}{X}_{n,k}^{\prime \prime })\right|>\frac{\epsilon b_n}{2}\right\}$$

$$⩽\frac{2^q}{{\epsilon }^q{b}_n^q}\mathbb{E}{\left|\sum _{k=1}^n(X_{n,k}^{\prime }-\mathbb{E}{X}_{n,k}^{\prime })\right|}^q+\frac{2}{\epsilon b_n}\mathbb{E}\left|\sum _{k=1}^n(X_{n,k}^{\prime \prime }-\mathbb{E}{X}_{n,k}^{\prime \prime })\right|$$

$$⩽\frac{2^q{\beta }_n}{{\epsilon }^q{b}_n^q}\sum _{k=1}^n\mathbb{E}|X_{n,k}^{\prime }{|}^q+\frac{2^q{\xi }_n}{{\epsilon }^q{b}_n^q}{\left(\sum _{k=1}^n\mathbb{E}|X_{n,k}^{\prime }{|}^2\right)}^{q/2}+\frac{4}{\epsilon b_n}\sum _{k=1}^n\mathbb{E}|X_{n,k}^{\prime \prime }|$$

$$⩽\frac{2^{2q-1}{\beta }_n}{{\epsilon }^q{b}_n^q}\sum _{k=1}^n\left[\mathbb{E}|X_{n,k}{|}^q{I}_{\left\{|X_{n,k}|⩽b_n\right\}}+b_n^q\mathbb{P}\left\{|X_{n,k}|>b_n\right\}\right]+\frac{4}{\epsilon b_n}\sum _{k=1}^n\mathbb{E}|X_{n,k}|I_{\left\{|X_{n,k}|>b_n\right\}}$$

$$+\frac{2^{3q/2}{\xi }_n}{{\epsilon }^q{b}_n^q}{\left(\sum _{k=1}^n\mathbb{E}{X}_{n,k}^2{I}_{\left\{|X_{n,k}|⩽b_n\right\}}+b_n^2\sum _{k=1}^n\mathbb{P}\left\{|X_{n,k}|>b_n\right\}\right)}^{q/2}$$

Full text

Key words: convergence of series of moments, dependent random variables

2010 Mathematics Subject Classification: 60F15

1 Introduction

In [5], Li and Spătaru proved the following statement: if $\{X_{n},\,n\geqslant 1\}$ is a sequence of independent and identically distributed (i.i.d.) random variables with $\mathbb{E}\,X_{1}=0$ and $p>0$, $0<q<2$, $r\geqslant 1$ are such that $qr\geqslant 1$, then

[TABLE]

is equivalent to

[TABLE]

A few years later, Chen and Wang [2] showed that, letting $p>0$, $\{X_{n},\,n\geqslant 1\}$ be a random sequence and $\{b_{n}\}$, $\{c_{n}\}$ be sequences of positive real numbers,

[TABLE]

and

[TABLE]

are equivalent. Hence, putting $x_{+}=\max(x,0)$, Li and Spătaru’s result can be restated as: if $\{X_{n},\,n\geqslant 1\}$ is a sequence of i.i.d. random variables with $\mathbb{E}\,X_{1}=0$ and $p>0$, $0<q<2$, $r\geqslant 1$ are such that $qr\geqslant 1$, then (1.1) is equivalent to

[TABLE]

The extension of (1.2) to arrays of (dependent) random variables has been emerged in literature over the last years (see [9], [11], [12], [13] or more recently [14]). Our purpose in this paper is to give general sufficient conditions to obtain

[TABLE]

when $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ is a triangular array of random variables and $\{b_{n}\}$, $\{c_{n}\}$ are sequences of positive constants. Namely, we shall assume that row sums of a suitable truncated triangular array of random variables satisfies classical moment inequalities, scilicet, a von Bahr-Esseen type inequality [10] or a Rosenthal type inequality (see, for instance, [8] page $59$). These are general assumptions which cover well-known dependent structures, particularly extended negatively dependence or pairwise negatively quadrant dependence (see Lemma 3.6 in last section).

In the sequel, we shall denote the indicator random variable of an event $A$ by $I_{A}$ and, for each $t>0$, we shall define also the function $g_{t}(x)=\max(\min(x,t),-t)$ which describes the truncation at level $t$.

2 Main results

In our first two statements, we shall establish the convergence of series (1.3) by assuming that, for any $t>0$, the (truncated) array of random variables $\left\{g_{t}(X_{n,k}),\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ satisfies a von Bahr-Esseen type inequality, i.e. there is a sequence of positive numbers $\{\alpha_{n}\}$ such that for some $q>1$,

[TABLE]

for all $n\geqslant 1$ and $t>0$.

Theorem 2.1**.**

*Let $p>1$, $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert^{p}<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.1) for a $q>p$ and some sequence $\{\alpha_{n}\}$ of positive numbers. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(a)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\alpha_{n}c_{n}b_{n}^{-q}\int_{0}^{b_{n}^{q}}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{q}>t\right\}\,\mathrm{d}t<\infty$, *

(b)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}\mathbb{E}\,\lvert X_{n,k}\rvert I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}/b_{n}<\infty$, *

(c)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}(1+\alpha_{n})c_{n}b_{n}^{-p}\int_{b_{n}^{p}}^{\infty}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{p}>t\right\}\,\mathrm{d}t<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

Fixing $\varepsilon>0$, we have

[TABLE]

Defining $X_{n,k}^{\prime}:=g_{b_{n}}(X_{n,k})$ and $X_{n,k}^{\prime\prime}=X_{n,k}-X_{n,k}^{\prime}$, Chebyshev inequality and assumption (2.1) entail

[TABLE]

Setting $Y_{n,k}^{\prime}:=g_{t^{1/p}}(X_{n,k})$ and $Y_{n,k}^{\prime\prime}=X_{n,k}-Y_{n,k}^{\prime}$, it follows

[TABLE]

Hence,

[TABLE]

On the other hand, \big{\lvert}Y_{n,k}^{\prime\prime}\big{\rvert}\leqslant\left\lvert X_{n,k}\right\rvert I_{\left\{\left\lvert X_{n,k}\right\rvert>t^{1/p}\right\}}, we obtain for every $p>1$

[TABLE]

Thus, by gathering (2.2), (2.3), (2.4) and (2.5) we get

[TABLE]

according to assumptions (a), (b) and (c). The proof is complete. ∎

Theorem 2.2**.**

*Let $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.1) for a $q>1$ and some sequence $\{\alpha_{n}\}$ of positive numbers. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(a)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\alpha_{n}c_{n}b_{n}^{-q}\int_{0}^{b_{n}^{q}}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{q}>t\right\}\,\mathrm{d}t<\infty$, *

(b)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}\mathbb{E}\,\lvert X_{n,k}\rvert I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}/b_{n}<\infty$, *

(c)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}(\alpha_{n}c_{n}/b_{n})\int_{b_{n}}^{\infty}\mathbb{P}\left\{\lvert X_{n,k}\rvert>t\right\}\,\mathrm{d}t<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

All steps in the proof of Theorem 2.1 remains true for $p=1$ except the upper bound (2.5). Supposing $Y_{n,k}^{\prime}:=g_{t}(X_{n,k})$ and $Y_{n,k}^{\prime\prime}=X_{n,k}-Y_{n,k}^{\prime}$ we have, for any $t\geqslant b_{n}$,

[TABLE]

Hence,

[TABLE]

and

[TABLE]

by employing (2.2), (2.4) with $p=1$ and (2.3), (2.7). The thesis is established. ∎

The next two results, give us conditions for the convergence of (1.3) under the assumption that, for every $t>0$, the array of random variables $\left\{g_{t}(X_{n,k}),\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ satisfies a Rosenthal type inequality. Specifically, we shall admit that there are sequences of positive numbers $\{\beta_{n}\}$ and $\{\xi_{n}\}$ such that for some $q>2$,

[TABLE]

for all $n\geqslant 1$ and $t>0$.

Theorem 2.3**.**

*Let $p>1$, $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert^{p}<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.8) for a $q>\max\{p,2\}$ and some sequences $\{\beta_{n}\}$, $\{\xi_{n}\}$ of positive numbers. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(a)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\beta_{n}c_{n}b_{n}^{-q}\int_{0}^{b_{n}^{q}}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{q}>t\right\}\,\mathrm{d}t<\infty$, *

(b)* $\sum_{n=1}^{\infty}\xi_{n}c_{n}b_{n}^{-p}\int_{0}^{b_{n}^{p-q}}\left(\sum_{k=1}^{n}\int_{0}^{t^{2/(p-q)}}\mathbb{P}\{X_{n,k}^{2}>s\}\mathrm{d}s\right)^{q/2}\mathrm{d}t<\infty$, *

(c)* $\sum_{n=1}^{\infty}\xi_{n}c_{n}b_{n}^{-q}\left(\sum_{k=1}^{n}\int_{0}^{b_{n}^{2}}\mathbb{P}\{X_{n,k}^{2}>t\}\,\mathrm{d}t\right)^{q/2}<\infty$, *

(d)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}\mathbb{E}\,\lvert X_{n,k}\rvert I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}/b_{n}<\infty$, *

(e)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}(1+\beta_{n})c_{n}b_{n}^{-p}\int_{b_{n}^{p}}^{\infty}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{p}>t\right\}\,\mathrm{d}t<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

The proof follows in exactly the same manner as the proof of Theorem 2.1 except for upper bounds (2.3) and (2.4) which must be replaced. Letting $X_{n,k}^{\prime}:=g_{b_{n}}(X_{n,k})$ and $X_{n,k}^{\prime\prime}=X_{n,k}-X_{n,k}^{\prime}$, assumption (2.8) ensures

[TABLE]

On the other hand, considering $Y_{n,k}^{\prime}:=g_{t^{1/p}}(X_{n,k})$ we have

[TABLE]

Employing (2.2), (2.5), (2.9) and (2.10) as in (2.6) the conclusion follows. The proof is complete. ∎

Theorem 2.4**.**

*Let $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.8) for a $q>2$ and some sequences $\{\beta_{n}\}$, $\{\xi_{n}\}$ of positive numbers. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(a)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\beta_{n}c_{n}b_{n}^{-q}\int_{0}^{b_{n}^{q}}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{q}>t\right\}\,\mathrm{d}t<\infty$, *

(b)* $\sum_{n=1}^{\infty}(\xi_{n}c_{n}/b_{n})\int_{0}^{b_{n}^{1-q}}\left(\sum_{k=1}^{n}\int_{0}^{t^{2/(1-q)}}\mathbb{P}\{X_{n,k}^{2}>s\}\mathrm{d}s\right)^{q/2}\mathrm{d}t<\infty$ *

(c)* $\sum_{n=1}^{\infty}\xi_{n}c_{n}b_{n}^{-q}\left(\sum_{k=1}^{n}\int_{0}^{b_{n}^{2}}\mathbb{P}\{X_{n,k}^{2}>t\}\,\mathrm{d}t\right)^{q/2}<\infty$, *

(d)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}\mathbb{E}\,\lvert X_{n,k}\rvert I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}/b_{n}<\infty$, *

(e)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}(\beta_{n}c_{n}/b_{n})\int_{b_{n}}^{\infty}\mathbb{P}\left\{\lvert X_{n,k}\rvert>t\right\}\,\mathrm{d}t<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

The thesis is a consequence of (2.2), (2.10) with $p=1$ and (2.7), (2.9). ∎

Remark 2.5*.*

Notice that if $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ is an array of row-wise extended negatively dependent random variables with dominating sequence $\{M_{n},n\geqslant 1\}$ (see [6]), then (2.8) holds with $q\geqslant 2$ and $\beta_{n}=\xi_{n}=C(q)(1+M_{n})$ with $C(q)$ a positive constant depending only on $q$ (see Lemma 2 of [6]); further, (2.1) still holds for these dependent structures with $1\leqslant q\leqslant 2$ and $\alpha_{n}=C(q)(1+M_{n})$, where $C(q)>0$ depends only on $q$.

Supposing $0<p\leqslant 1$, $\varepsilon>0$ and $b_{n}$ a real sequence of positive numbers, Lemma 2.1 of [9] and elementary inequality $(x+y)^{p}\leqslant x^{p}+y^{p}$, $x,y\geqslant 0$ lead us to

[TABLE]

By taking $q>p$, we obtain

[TABLE]

which yields

[TABLE]

Hence, we can still announce the result hereinafter whose proof follows from inequality (2.11); we omit the details.

Theorem 2.6**.**

*Let $0<p<1$, $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert^{p}<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(a)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-q}\mathbb{E}\lvert X_{n,k}\rvert^{q}I_{\left\{\lvert X_{n,k}\rvert\leqslant b_{n}\right\}}<\infty$ for some $p<q\leqslant 1$, *

(b)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-p}\mathbb{E}\,\lvert X_{n,k}\rvert^{p}I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Remark 2.7*.*

Under the assumptions of Theorem 2.1 (or Theorem 2.3) we obviously have, for any $0<r\leqslant p$,

[TABLE]

for all $\varepsilon>0$, because

[TABLE]

and

[TABLE]

In the same way, (2.12) holds for every $0<r\leqslant 1$ whenever the assumptions of Theorem 2.2 (or Theorem 2.4) are met.

3 Applications

It is straightforward to see that

[TABLE]

and

[TABLE]

for any $p,q,u>0$. Thus, if $\{\alpha_{n}\}$ is a constant sequence then both Theorems 2.1 and 2.2 can be gathered in the following result.

Corollary 3.1**.**

*Let $p\geqslant 1$, $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert^{p}<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.1) for a $q>p$ and some constant sequence $\{\alpha_{n}\}$. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(i)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-q}\mathbb{E}\,\lvert X_{n,k}\rvert^{q}I_{\left\{\lvert X_{n,k}\rvert\leqslant b_{n}\right\}}<\infty$, *

(ii)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-p}\mathbb{E}\,\lvert X_{n,k}\rvert^{p}I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

Since $\{\alpha_{n}\}$ is a constant sequence, (ii) ensures assumption (c) of Theorems 2.1 and 2.2 via (3.1). According to (3.3), (i) and (ii) together guarantee assumption (a) of Theorems 2.1 and 2.2. Finally, assumption (b) of Theorems 2.1 and 2.2 follows from (ii) by noting that

[TABLE]

Hence, Corollary 3.1 is proved. ∎

Similarly, we can also join Theorems 2.3 and 2.4 when sequences $\{\beta_{n}\}$ and $\{\xi_{n}\}$ are constant.

Corollary 3.2**.**

*Let $p\geqslant 1$, $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of random variables satisfying $\mathbb{E}\lvert X_{n,k}\rvert^{p}<\infty$ for each $1\leqslant k\leqslant n$ and all $n\geqslant 1$, and verifying (2.8) for a $q>\max\{p,2\}$ and some constant sequences $\{\beta_{n}\}$, $\{\xi_{n}\}$. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(i)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-q}\mathbb{E}\,\lvert X_{n,k}\rvert^{q}I_{\left\{\lvert X_{n,k}\rvert\leqslant b_{n}\right\}}<\infty$, *

(ii)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}c_{n}b_{n}^{-p}\mathbb{E}\,\lvert X_{n,k}\rvert^{p}I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}<\infty$, *

(iii)* $\sum_{n=1}^{\infty}c_{n}b_{n}^{-pq/2}\left(\sum_{k=1}^{n}\mathbb{E}\,\lvert X_{n,k}\rvert^{p}I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}\right)^{q/2}<\infty$, *

(iv)* $\sum_{n=1}^{\infty}c_{n}b_{n}^{-q}\left(\sum_{k=1}^{n}\mathbb{E}\,X_{n,k}^{2}I_{\left\{\lvert X_{n,k}\rvert\leqslant b_{n}\right\}}\right)^{q/2}<\infty$, *

then

[TABLE]

for all $\varepsilon>0$.

Proof.

From (iii) and (iv), we have that conditions (b), (c) of Theorems 2.3 and 2.4 are verified since

[TABLE]

and

[TABLE]

The remaining assumptions of Theorems 2.3 and 2.4 follow from (i), (ii) as in the proof of Corollary 3.1. The proof is complete. ∎

Let $\{\Psi_{n,k}(x),\,1\leqslant k\leqslant n,n\geqslant 1\}$ be an array of functions defined on $[0,\infty)$ satisfying for all $n\geqslant 1$ and every $1\leqslant k\leqslant n$,

[TABLE]

for some $1\leqslant p<q$.

Corollary 3.3**.**

*Let $\{\Psi_{n,k}(x),\,1\leqslant k\leqslant n,n\geqslant 1\}$ be an array of functions defined on $[0,\infty)$ verifying (3.4) for some $1\leqslant p<q\leqslant 2$, and $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of zero-mean random variables satisfying (2.1) for such $q$ and some constant sequence $\alpha_{n}$. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(1)* $\sum_{n=1}^{\infty}c_{n}\sum_{k=1}^{n}\mathbb{E}\Psi_{n,k}(\lvert X_{n,k}\rvert)/\Psi_{n,k}(b_{n})<\infty$, *

then $\sum_{n=1}^{\infty}c_{n}\mathbb{E}\left(\left\lvert\sum_{k=1}^{n}X_{n,k}\right\rvert/b_{n}-\varepsilon\right)_{+}^{p}<\infty$ for all $\varepsilon>0$.

Proof.

From $\Psi_{n,k}(t)/t^{q}\downarrow$ as $0<t\uparrow$, it follows

[TABLE]

for all $1\leqslant k\leqslant n$ and $n\geqslant 1$. On the other hand, $\Psi_{n,k}(t)/t^{p}\uparrow$ as $0<t\uparrow$ entails

[TABLE]

for each $1\leqslant k\leqslant n$ and $n\geqslant 1$. Hence, (3.5) and (3.7) yield

[TABLE]

for any $1\leqslant k\leqslant n$ and $n\geqslant 1$, which assures assumption (i) of Corollary 3.1. Moreover, (3.6) and (3.8) imply

[TABLE]

for every $1\leqslant k\leqslant n$ and $n\geqslant 1$. Thus, assumption (ii) of Corollary 3.1 holds via (3.9) and (1). The proof is complete. ∎

Corollary 3.4**.**

*Let $\{\Psi_{n,k}(x),\,1\leqslant k\leqslant n,n\geqslant 1\}$ be an array of functions defined on $[0,\infty)$ verifying (3.4) for some $p\geqslant 1$ and $q>\max\{2,p\}$, and $\left\{X_{n,k},\,1\leqslant k\leqslant n,n\geqslant 1\right\}$ be an array of zero-mean random variables satisfying (2.8) for such $q$ and some constant sequences $\beta_{n}$, $\xi_{n}$. If $\{b_{n}\}$, $\{c_{n}\}$ are real sequences of positive numbers such that

(1)* $\sum_{n=1}^{\infty}c_{n}\sum_{k=1}^{n}\mathbb{E}\Psi_{n,k}(\lvert X_{n,k}\rvert)/\Psi_{n,k}(b_{n})<\infty$, *

(2)* $\sum_{n=1}^{\infty}c_{n}\left(\sum_{k=1}^{n}\mathbb{E}\,X_{n,k}^{2}I_{\left\{\lvert X_{n,k}\rvert\leqslant b_{n}\right\}}/b_{n}^{2}\right)^{q/2}<\infty$, *

(3)* $\sum_{n=1}^{\infty}c_{n}\left[\sum_{k=1}^{n}\mathbb{E}\Psi_{n,k}(\lvert X_{n,k}\rvert)/\Psi_{n,k}(b_{n})\right]^{q/2}<\infty$, *

then $\sum_{n=1}^{\infty}c_{n}\mathbb{E}\left(\left\lvert\sum_{k=1}^{n}X_{n,k}\right\rvert/b_{n}-\varepsilon\right)_{+}^{p}<\infty$.

Proof.

The thesis is a consequence of Corollary 3.2 by arguing as in the proof of Corollary 3.3. ∎

Remark 3.5*.*

We observe that Theorem 3 of [12] can be obtained via Corollaries 3.3 and 3.4 by taking $c_{n}=1$ for all $n\geqslant 1$ and $\Psi_{n,k}(x)$ not depending on $n,k$ satisfying $\Psi_{n,k}(0)=0$; indeed, for such sequence $c_{n}$, the assumption $\sum_{n=1}^{\infty}c_{n}\left[\sum_{k=1}^{n}\mathbb{E}\Psi_{n,k}(\lvert X_{n,k}\rvert)/\Psi_{n,k}(b_{n})\right]^{q/2}<\infty$ can be dropped in Corollary 3.4.

The lemma below gives us a von Bahr-Esseen type inequality for row-wise pairwise negative quadrant dependent (NQD) triangular arrays (see, for instance, [7]). The proof can be performed as in Theorem 2.1 of [1] by employing the truncation $X_{n,k}^{\prime}=g_{x^{1/r}}(X_{n,k})$, $1<r<2$ and $X_{n,k}^{\prime\prime}=X_{n,k}-X_{n,k}^{\prime}$, being thus omitted.

Lemma 3.6**.**

Let $1\leqslant r\leqslant 2$ and $\{X_{n,k},\,1\leqslant k\leqslant n,\,n\geqslant 1\}$ be a triangular array of zero-mean row-wise pairwise NQD random variables such that $\mathbb{E}\,\lvert X_{n,k}\rvert^{r}<\infty$ for all $n\geqslant 1$ and any $1\leqslant k\leqslant n$. Then

[TABLE]

where $C(r)>0$ depends only on $r$.

Remark 3.7*.*

It is worthy to note that using Lemma 3.6 in Theorems 2.1 and 2.2, we can extend Theorem 3.7 of [1] to sequences $\{X_{n},\,n\geqslant 1\}$ of pairwise NQD and identically distributed random variables, by admitting $X_{n,k}=X_{k}$, $p=r$, $c_{n}=n^{t-2}$ and $b_{n}=n^{1/\rho}$ $(0<\rho<2)$ with $1\leqslant r\leqslant 2$, $t\geqslant 1$, and $t\rho<2$.

Remark 3.8*.*

Let us point out that all statements presented throughout can be properly extended without effort to general arrays $\{X_{n,j},\,1\leqslant j\leqslant k_{n},\,n\geqslant 1\}$, where $\{k_{n}\}$ is a sequence of positive integers such that $k_{n}\rightarrow\infty$ as $n\rightarrow\infty$. Considering also an array $\{\Psi_{n,j}(x),\,1\leqslant j\leqslant k_{n},n\geqslant 1\}$ of functions defined on $[0,\infty)$ satisfying (3.4) for every $1\leqslant j\leqslant k_{n}$ and all $n\geqslant 1$, we conclude from Lemma 3.6 that our Corollary 3.3 extends Theorem 1.1 of [14].

Corollary 3.9**.**

*Let $1<r<2$ and $\{X_{n,k},\,1\leqslant k\leqslant n,\,n\geqslant 1\}$ be a triangular array of row-wise pairwise NQD random variables such that $\mathbb{E}\,\lvert X_{n,k}\rvert^{r}<\infty$ for all $n\geqslant 1$ and any $1\leqslant k\leqslant n$. If $1\leqslant p<r$ and $\{b_{n}\}$ is a real sequence of positive constants satisfying,

(1)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}b_{n}^{-r}\int_{0}^{b_{n}^{r}}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{r}>t\right\}\,\mathrm{d}t<\infty$, *

(2)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\mathbb{E}\,\lvert X_{n,k}\rvert I_{\left\{\lvert X_{n,k}\rvert>b_{n}\right\}}/b_{n}<\infty$, *

(3)* $\sum_{n=1}^{\infty}\sum_{k=1}^{n}b_{n}^{-p}\int_{b_{n}^{p}}^{\infty}\mathbb{P}\left\{\lvert X_{n,k}\rvert^{p}>t\right\}\,\mathrm{d}t<\infty$, *

then $\sum_{n=1}^{\infty}\mathbb{E}\left[\left\lvert\sum_{k=1}^{n}(X_{n,k}-\mathbb{E}\,X_{n,k})\right\rvert/b_{n}-\varepsilon\right]_{+}^{p}<\infty$ for all $\varepsilon>0$.

Proof.

From previous Lemma 3.6 we obtain (2.1) with $q=r$ and $\alpha_{n}=C(r)$. The thesis follows from Theorems 2.1 and 2.2 by taking $c_{n}=1$ for all $n\geqslant 1$. ∎

4 Final comments

In 1947, Hsu and Robbins [4] introduced the concept of complete convergence (see also [3] for a survey). By taking $c_{n}=1$ for all $n\geqslant 1$ in (1.3), we obtain that $\left\lvert\sum_{k=1}^{n}(X_{n,k}-\mathbb{E}\,X_{n,k})\right\rvert/b_{n}$ converges completely to zero: indeed, setting $A_{n}(\varepsilon):=\left\{\omega\colon\left\lvert\sum_{k=1}^{n}(X_{n,k}-\mathbb{E}\,X_{n,k})\right\rvert/b_{n}-\varepsilon\geqslant 0\right\}$, we have, for each $\delta>0$,

[TABLE]

Furthermore, by using the elementary inequality $\lvert x\rvert^{p}\leqslant\max(1,2^{p-1})\left[(\lvert x\rvert-\varepsilon)_{+}^{p}+\varepsilon^{p}\right]$ for every real number $x$ and any $p,\varepsilon>0$, it follows

[TABLE]

so that, our statements also guarantee the convergence in mean of order $p$ (to zero) for triangular arrays of random variables under the considered assumptions.

Acknowledgements

This work is a contribution to the Project UIDB/04035/2020, funded by FCT - Fundação para a Ciência e a Tecnologia, Portugal.