On complete convergence for weighted sums of martingale-difference random fields

Abstract

Let $\{a_{\mathbf{n},\mathbf{i}},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ be an array of real numbers, and let $\{X_{\mathbf{i}},\mathbf{i}\in Z_{+}^d\}$ be the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying $E(E(X|{\mathcal{G}}_k)|{\mathcal{G}}_{\mathbf{m}})=E(X|{\mathcal{G}}_{\mathbf{k}\wedge \mathbf{m}})$ a.s., where $\mathbf{k}\wedge \mathbf{m}$ denotes componentwise minimum, $\{{\mathcal{G}}_{\mathbf{k}},\mathbf{k}\in Z_{+}^d\}$ is a family of σ-algebras such that $\mathrm{\forall }\mathbf{k}\le \mathbf{n}$, ${\mathcal{G}}_{\mathbf{k}}\subset {\mathcal{G}}_{\mathbf{n}}\subset \mathcal{G}$, and X is any integrable random variable defined on the initial probability space. The aim of this paper is to obtain some results concerning complete convergence of weighted sums ${\sum }_{\mathbf{i}\le \mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}$.

MSC:60F05, 60F15.

1 Introduction

The concept of complete convergence for sums of independent and identically distributed random variables was introduced by Hsu and Robbins [1] as follows: A sequence of random variables $\{X_n\}$ is said to be completely to a constant c if

$$\sum _{n=1}^{\mathrm{\infty }}P(|X_n-c|>ϵ)<\mathrm{\infty }\text{for all }ϵ>0.$$

This result has been generalized and extended to the random fields $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ of random variables. For example, Fazekas and Tómács [2] and Czerebak-Mrozowicz et al. [3] for fields of pairwise independent random variables, and Gut and Stadtmüller [4] for random fields of i.i.d. random variables.

Let $Z_{+}$ be the set of positive integers. For fixed $d\in Z_{+}$, set $Z_{+}^d=\{\mathbf{n}=(n_1,n_2,\dots ,n_d):n_i\in Z_{+},i=1,2,\dots ,d\}$ with coordinatewise partial order, ≤, i.e., for $\mathbf{m}=(m_1,m_2,\dots ,m_d),\mathbf{n}=(n_1,n_2,\dots ,n_d)\in Z_{+}^d$, $\mathbf{m}\le \mathbf{n}$ if and only if $m_i\le n_i$, $i=1,2,\dots ,d$. For $\mathbf{n}=(n_1,n_2,\dots ,n_d)\in Z_{+}^d$, let $|\mathbf{n}|={\prod }_{i=1}^d{n}_i$. For a field $\{a_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ of real numbers, the limit superior is defined by ${inf}_{r\ge 1}{sup}_{|\mathbf{n}|\ge r}{a}_{\mathbf{n}}$ and is denoted by ${lim\hspace{0.17em}sup}_{|\mathbf{n}|\to \mathrm{\infty }}{a}_{\mathbf{n}}$.

Note that $|\mathbf{n}|\to \mathrm{\infty }$ is equivalent to $max\{n_1,n_2,\dots ,n_d\}\to \mathrm{\infty }$, which is weaker than the condition $min\{n_1,n_2,\dots ,n_d\}\to \mathrm{\infty }$ when $d\ge 2$.

Let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be a field of random variables, and let $\{a_{\mathbf{n},\mathbf{k}},\mathbf{n}\in Z_{+}^d,\mathbf{k}\le \mathbf{n}\}$ be an array of real numbers. The weighted sums ${\sum }_{\mathbf{k}\le \mathbf{n}}{a}_{\mathbf{n},\mathbf{k}}{X}_{\mathbf{k}}$ can play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles and Bhaskara Rao [5]) and M-estimates (see Rao and Zhao [6]) in linear models, the nonparametric regression estimators (see Priestley and Chao [7]), etc. So, the study of the limiting behavior of the weighted sums is very important and significant (see Chen and Hao [8]).

Now, we consider the notion of martingale differences. Let $\{{\mathcal{G}}_{\mathbf{k}},\mathbf{k}\in Z_{+}^d\}$ be a family of σ-algebras such that

$${\mathcal{G}}_{\mathbf{k}}\subset {\mathcal{G}}_{\mathbf{n}}\subset \mathcal{G},\mathrm{\forall }\mathbf{k}\le \mathbf{n},$$

and for any integrable random variable X defined on the initial probability space,

$$E(E(X|{\mathcal{G}}_{\mathbf{k}})|{\mathcal{G}}_{\mathbf{m}})=E(X|{\mathcal{G}}_{\mathbf{k}\wedge \mathbf{m}})\text{a.s.},$$

(1.1)

where $\mathbf{k}\wedge \mathbf{m}$ denotes the componentwise minimum.

An $\{{\mathcal{G}}_{\mathbf{k}},\mathbf{k}\in Z_{+}^d\}$-adapted, integrable process $\{Y_{\mathbf{k}},\mathbf{k}\in Z_{+}^d\}$ is called a martingale if and only if

$$E(Y_{\mathbf{n}}|{\mathcal{G}}_{\mathbf{m}})=Y_{\mathbf{m}\wedge \mathbf{n}}\text{a.s.}$$

Let us observe that for martingale $\{(Y_{\mathbf{n}},{\mathcal{G}}_{\mathbf{n}}),\mathbf{n}\in Z_{+}^d\}$, the random variables

$$X_{\mathbf{n}}=\sum _{\mathbf{a}\in {\{0,1\}}^d}{(-1)}^{{\sum }_{\mathbf{i}=1}^d{a}_i}{Y}_{\mathbf{n}-\mathbf{a}},$$

where $\mathbf{a}=(a_1,a_2,\dots ,a_d)$ and $\mathbf{n}\in Z_{+}^d$, are martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ (see Kuczmaszewska and Lagodowski [9]).

For the results concerning complete convergence for martingale arrays obtained in the one-dimensional case, we refer to Lagodowski and Rychlik [10], Elton [11], Lesigne and Volny [12], Stoica [13] and Ghosal and Chandra [14]. Recently, complete convergence for martingale difference random fields was proved by Kuczmaszewska and Lagodowski [9].

The aim of this paper is to obtain some results concerning complete convergence of weighted sums ${\sum }_{\mathbf{i}\le \mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}$, where $\{a_{\mathbf{n},\mathbf{i}},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ is an array of real numbers, and $\{X_{\mathbf{i}},\mathbf{i}\in Z_{+}^d\}$ is the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1).

2 Results

The following moment maximal inequality provides us a useful tool to prove the main results of this section (see Kuczmaszewska and Lagodowski [9]).

Lemma 2.1 Let $\{(Y_{\mathbf{n}},{\mathcal{G}}_{\mathbf{n}}),\mathbf{n}\in Z_{+}^d\}$ be a martingale, and let $\{(X_{\mathbf{n}},{\mathcal{G}}_{\mathbf{n}}),\mathbf{n}\in Z_{+}^d\}$ be the martingale differences corresponding to it. Let $q>1$. There exists a finite and positive constant C depending only on q and d such that

$$E\left(\underset{\mathbf{k}\le \mathbf{n}}{max}{|Y_{\mathbf{k}}|}^q\right)\le CE{\left(\sum _{\mathbf{k}\le \mathbf{n}}{X}_{\mathbf{k}}^2\right)}^{q/2}.$$

(2.1)

Let us denote ${\mathcal{G}}_{\mathbf{i}}^{\ast }=\sigma \{{\mathcal{G}}_{\mathbf{j}}:\mathbf{j}<\mathbf{i}\}$. Now, we are ready to formulate the next result.

Theorem 2.2 Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ be an array of real numbers, and let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1). For $\alpha p>1$, $p>1$ and $\alpha >\frac{1}{2}$, we assume that

• (i) ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}{\sum }_{\mathbf{i}\le \mathbf{n}}P\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{|\mathbf{n}|}^{\alpha }\}<\mathrm{\infty }$,

• (ii) ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha (p-q)-3+q/2}{\sum }_{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{q}E({|X_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }])<\mathrm{\infty }$ for $q\ge 2$,

(ii)′ ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha (p-q)-2}{\sum }_{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{q}E({|X_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }])<\mathrm{\infty }$ for $1<q<2$ and

• (iii) ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{{max}_{\mathbf{j}\le \mathbf{n}}|{\sum }_{\mathbf{i}\le \mathbf{j}}E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })|>ϵ{|\mathbf{n}|}^{\alpha }\}<\mathrm{\infty }$ for all $ϵ>0$.

Then we have

$$\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}|S_{\mathbf{j}}|>ϵ{|\mathbf{n}|}^{\alpha }\}<\mathrm{\infty }\mathit{\text{for all }}ϵ>0,$$

(2.2)

where $S_{\mathbf{n}}={\sum }_{\mathbf{1}\le \mathbf{i}\le \mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}$.

Proof Let us notice that the series ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}$ is finite, then (2.2) always holds. Therefore, we consider only the case such that ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}$ is divergent. Let $X_{\mathbf{n},\mathbf{i}}=X_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]$, $X_{\mathbf{n},\mathbf{i}}^{\ast }=X_{\mathbf{n},\mathbf{i}}-E(X_{\mathbf{n},\mathbf{i}}|{\mathcal{G}}_{\mathbf{i}}^{\ast })$ and $S_{\mathbf{n},\mathbf{j}}^{\ast }={\sum }_{\mathbf{i}\le \mathbf{j}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}^{\ast }$.

Then

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}|S_{\mathbf{j}}|>ϵ{|\mathbf{n}|}^{\alpha }\}\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{|\mathbf{n}|}^{\alpha }\}\hfill \\ +\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|\sum _{\mathbf{i}\le \mathbf{j}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right|>ϵ{|\mathbf{n}|}^{\alpha }\}\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}\sum _{\mathbf{i}\le \mathbf{n}}P\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{|\mathbf{n}|}^{\alpha }\}+\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\left\{\underset{\mathbf{j}\le \mathbf{n}}{max}\right|\sum _{\mathbf{i}\le \mathbf{j}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\hfill \\ -E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })\left)\right|>\frac{ϵ}{2}{|\mathbf{n}|}^{\alpha }\}\hfill \\ +\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|\sum _{\mathbf{i}\le \mathbf{j}}E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })\right|>\frac{ϵ}{2}{|\mathbf{n}|}^{\alpha }\}\hfill \\ =I_1+I_2+I_3.\hfill \end{array}$$

Clearly, $I_1<\mathrm{\infty }$ by (i), and $I_3<\mathrm{\infty }$ by (iii). It remains to prove that $I_2<\mathrm{\infty }$. Thus, the proof will be completed by proving that

$$\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|S_{\mathbf{n},\mathbf{j}}^{\ast }\right|>ϵ{|\mathbf{n}|}^{\alpha }\}<\mathrm{\infty }.$$

To prove it, we first observe that $\{(S_{\mathbf{n},\mathbf{j}}^{\ast },{\mathcal{G}}_{\mathbf{j}}),\mathbf{j}\le \mathbf{n}\}$ is a martingale. In fact, if $\mathbf{i}>\mathbf{j}$, then ${\mathcal{G}}_{\mathbf{i}\wedge \mathbf{j}}\subset {\mathcal{G}}_{\mathbf{i}}^{\ast }$ and by (1.1), we have

$$\begin{array}{rcl}E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}^{\ast }|{\mathcal{G}}_{\mathbf{j}})& =& E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}-E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}|{\mathcal{G}}_{\mathbf{i}}^{\ast })|{\mathcal{G}}_{\mathbf{i}})\\ =& E(E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}-E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}|{\mathcal{G}}_{\mathbf{i}}^{\ast })|{\mathcal{G}}_{\mathbf{i}})|{\mathcal{G}}_{\mathbf{j}})\\ =& E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}-E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}|{\mathcal{G}}_{\mathbf{i}}^{\ast })|{\mathcal{G}}_{\mathbf{i}\wedge \mathbf{j}})\\ =& 0.\end{array}$$

Then, by the Markov inequality and Lemma 2.1, there exists some constant C such that

$$\begin{array}{rcl}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|S_{\mathbf{n},\mathbf{j}}^{\ast }\right|>ϵ{|\mathbf{n}|}^{\alpha }\}& \le & C\frac{E({max}_{\mathbf{j}\le \mathbf{n}}{|S_{\mathbf{n},\mathbf{j}}^{\ast }|}^q)}{{|\mathbf{n}|}^{\alpha q}}\\ \le & \frac{C}{{|\mathbf{n}|}^{\alpha q}}E{\left(\sum _{\mathbf{i}\le \mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}^2{X}_{\mathbf{n},\mathbf{i}}^{\ast 2}\right)}^{q/2}=I_4.\end{array}$$

Case $q\ge 2$; we get

$$\begin{array}{rcl}{I}_4& \le & \frac{C}{{|\mathbf{n}|}^{\alpha q}}{|\mathbf{n}|}^{q/2-1}\sum _{\mathbf{i}\le \mathbf{n}}E{\left|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}^{\ast }\right|}^q\\ \le & C{|\mathbf{n}|}^{q/2-1-\alpha q}\sum _{\mathbf{i}\le \mathbf{n}}E\left({|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right).\end{array}$$

Note that the last estimation follows from the Jensen inequality. Thus, we have

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|S_{\mathbf{n},\mathbf{j}}^{\ast }\right|>ϵ{|\mathbf{n}|}^{\alpha }\}\hfill \\ \le C\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-3-q(\alpha -1/2)}\sum _{\mathbf{i}\le \mathbf{n}}E\left({|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right)<\mathrm{\infty }\hfill \end{array}$$

by assumption (ii).

Case $1<q<2$; we get

$$\begin{array}{rcl}{I}_4& \le & \frac{C}{{|\mathbf{n}|}^{\alpha q}}\sum _{\mathbf{i}\le \mathbf{n}}E{\left|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{n},\mathbf{i}}^{\ast }\right|}^q\\ \le & C{|\mathbf{n}|}^{-\alpha q}\sum _{\mathbf{i}\le \mathbf{n}}E\left({|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right).\end{array}$$

Therefore, for $1<q<2$, we obtain

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P\{\underset{\mathbf{j}\le \mathbf{n}}{max}\left|S_{\mathbf{n},\mathbf{j}}^{\ast }\right|>ϵ{|\mathbf{n}|}^{\alpha }\}\hfill \\ \le C\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha (p-q)-2}\sum _{\mathbf{i}\le \mathbf{n}}E\left({|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right)<\mathrm{\infty }\hfill \end{array}$$

by assumption (ii)′. Thus, $I_2<\mathrm{\infty }$ for all $q>1$, and the proof of Theorem 2.2 is complete. □

Corollary 2.3 Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ be an array of real numbers. Let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1), and $E{X}_{\mathbf{n}}=0$ for $\mathbf{n}\in Z_{+}^d$. Let $p\ge 1$, $\alpha >\frac{1}{2}$ and $\alpha p>1$. Assume that (i) and for some $q>1$, (ii) or (ii)′ hold respectively. If

$$\underset{\mathbf{j}\le \mathbf{n}}{max}\sum _{\mathbf{i}\le \mathbf{j}}E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })=o\left({|\mathbf{n}|}^{\alpha }\right),$$

(2.3)

then (2.2) holds.

Proof It is easy to see that (2.3) implies (iii). We omit details that prove it. □

The following corollary shows that assumption (iii) in Theorem 2.2 is natural, and in the case of independent random fields, it reduces to the known one.

Corollary 2.4 Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ be an array of real numbers. Let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be a field of independent random variables such that $E{X}_{\mathbf{n}}=0$ for $\mathbf{n}\in Z_{+}^d$. Let $p\ge 1$, $\alpha >\frac{1}{2}$ and $\alpha p>1$. Assume that (i) and for some $q>1$, (ii) or (ii)′ hold respectively. If

$$\frac{1}{{|\mathbf{n}|}^{\alpha }}\underset{\mathbf{j}\le \mathbf{n}}{max}\sum _{\mathbf{i}\le \mathbf{j}}E\left(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right)\to 0\mathit{\text{as }}|\mathbf{n}|\to \mathrm{\infty },$$

(2.4)

then (2.2) holds.

Proof Since $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ is a field of independent random variables, we have

$$\frac{1}{{|\mathbf{n}|}^{\alpha }}\underset{\mathbf{j}\le \mathbf{n}}{max}\sum _{\mathbf{i}\le \mathbf{j}}E(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })=\frac{1}{{|\mathbf{n}|}^{\alpha }}\underset{\mathbf{j}\le \mathbf{n}}{max}\sum _{\mathbf{i}\le \mathbf{j}}E\left(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right).$$

Now, it is easy to see that (2.4) implies (iii) of Theorem 2.2. Thus, by Theorem 2.2, result (2.2) follows. □

Remark Theorem 2.2 and Corollary 2.4 are extensions of Theorem 4.1 and Corollary 4.1 in Kuczmaszewska and Lagodowski [9] to the weighted sums case, respectively.

Corollary 2.5 Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{1}\le \mathbf{i}\le \mathbf{n}\}$ be an array of real numbers. Let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1) and $E{X}_{\mathbf{n}}=0$. Let $p\ge 1$, $\alpha >\frac{1}{2}$ and $\alpha p>1$ and $E{|X_{\mathbf{n}}|}^{1+{\lambda }_{\mathbf{n}}}<\mathrm{\infty }$ for ${\lambda }_{\mathbf{n}}$ with $0<{\lambda }_{\mathbf{n}}<1$ for $\mathbf{n}\in Z_{+}^d$. If

$$\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}{|\mathbf{n}|}^{-\alpha (1+{\lambda }_{\mathbf{n}})}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}E{|X_{\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}<\mathrm{\infty },$$

(2.5)

$$\underset{\mathbf{1}\le \mathbf{j}\le \mathbf{n}}{max}\sum _{\mathbf{i}\le \mathbf{j}}E(|a_{\mathbf{n},\mathbf{i}}{X}_i|I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]|{\mathcal{G}}_{\mathbf{i}}^{\ast })=o\left({|\mathbf{n}|}^{\alpha }\right),$$

(2.6)

then (2.2) holds.

Proof If the series ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}<\mathrm{\infty }$, then (2.2) always holds. Hence, we only consider the case ${\sum }_{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}=\mathrm{\infty }$. It follows from (2.5) that

$${|\mathbf{n}|}^{-\alpha (1+{\lambda }_{\mathbf{n}})}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}E{|X_i|}^{1+{\lambda }_{\mathbf{n}}}<1.$$

By (2.5) and the Markov inequality,

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}P(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{|\mathbf{n}|}^{\alpha })\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}{|\mathbf{n}|}^{-\alpha (1+{\lambda }_{\mathbf{n}})}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}E{|X_{\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}<\mathrm{\infty },\hfill \end{array}$$

(2.7)

which satisfies (i) of Theorem 2.2.

As the proof of Corollary 2.3, (2.6) implies (iii) of Theorem 2.2.

It remains to show that Theorem 2.2(ii) or (ii)′ is satisfied.

For $1<q<2$, take $1+{\lambda }_{\mathbf{n}}<q$. Then we have

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha (p-q)-2}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{q}E\left({|X_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {|\mathbf{n}|}^{\alpha }]\right)\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}{|\mathbf{n}|}^{-\alpha (1+{\lambda }_{\mathbf{n}})}{|\mathbf{n}|}^{-\alpha q+\alpha (1+{\lambda }_{\mathbf{n}})}{|\mathbf{n}|}^{\alpha q-\alpha (1+{\lambda }_{\mathbf{n}})}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}E{|X_{\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}\hfill \\ =\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}{|\mathbf{n}|}^{-\alpha (1+{\lambda }_{\mathbf{n}})}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{1+{\lambda }_{\mathbf{n}}}E{|X_i|}^{1+{\lambda }_{\mathbf{n}}}<\mathrm{\infty }\text{by (2.5)},\hfill \end{array}$$

which satisfies Theorem 2.2(ii)′. Hence, the proof is complete. □

Corollary 2.6 Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{1}\le \mathbf{i}\le \mathbf{n}\}$ be an array of real numbers, and let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1), $E{X}_{\mathbf{n}}=0$ and $E{|X_{\mathbf{n}}|}^p<\mathrm{\infty }$ for $1<p<2$. Let $\alpha >\frac{1}{2}$, $\alpha p>1$ and $1<p<2$. If

$$\sum _{\mathbf{1}\le \mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{p}E{|X_{\mathbf{i}}|}^p=O\left({|\mathbf{n}|}^{\delta }\right)\mathit{\text{for }}0<\delta <1,$$

(2.8)

and Theorem  2.2(iii) hold, then (2.2) holds.

Proof By (2.8) and the Markov inequality,

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}\sum _{\mathbf{i}\le \mathbf{n}}P(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{|\mathbf{n}|}^{\alpha })\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha p-2}\sum _{\mathbf{i}\le \mathbf{n}}\frac{{|a_{\mathbf{n},\mathbf{i}}|}^{p}E{|X_{\mathbf{i}}|}^p}{{|\mathbf{n}|}^{\alpha p}}\hfill \\ \le C\sum _{\mathbf{n}}{|\mathbf{n}|}^{-2+\delta }<\mathrm{\infty }.\hfill \end{array}$$

(2.9)

By taking $q<p$, we have

$$\begin{array}{c}\sum _{\mathbf{n}}{|\mathbf{n}|}^{\alpha (p-q)-2}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{q}E\left({|X_{\mathbf{i}}|}^{q}I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le ϵ{|\mathbf{n}|}^{\alpha }]\right)\hfill \\ \le \sum _{\mathbf{n}}{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^{p}E{|X_{\mathbf{i}}|}^p\hfill \\ \le C\sum _{\mathbf{n}}{|\mathbf{n}|}^{-2+\delta }<\mathrm{\infty }.\hfill \end{array}$$

(2.10)

Hence, by (2.9) and (2.10), conditions (i) and (ii)′ in Theorem 2.2 are satisfied, respectively.

To complete the proof, it is enough to note that by $E{X}_{\mathbf{n}}=0$ for $\mathbf{n}\in Z_{+}^d$ and by (2.8), we get for $\mathbf{j}\le \mathbf{n}$

$${|\mathbf{n}|}^{-\alpha }\sum _{\mathbf{i}\le \mathbf{j}}|a_{\mathbf{n},\mathbf{i}}|E|X_{\mathbf{i}}|I[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le ϵ{|\mathbf{n}|}^{\alpha }]\to 0\text{as }|\mathbf{n}|\to \mathrm{\infty }.$$

(2.11)

Hence, the proof is complete. □

Corollary 2.7 Let $\{X_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ be the martingale differences with respect to $\{{\mathcal{G}}_{\mathbf{n}},\mathbf{n}\in Z_{+}^d\}$ satisfying (1.1), let $E{X}_{\mathbf{n}}=0$ and $E{|X_{\mathbf{n}}|}^p<\mathrm{\infty }$ for $1<p<2$ and be stochastically dominated by a random variable X, i.e., there is a constant D such that $P(|X_{\mathbf{n}}|>x)\le DP(|X|>x)$ for all $x\ge 0$ and $\mathbf{n}\in Z_{+}^d$. Let $\{a_{\mathbf{n}},\mathbf{i},\mathbf{n}\in Z_{+}^d,\mathbf{i}\le \mathbf{n}\}$ be an array of real numbers satisfying

$$\sum _{\mathbf{i}\le \mathbf{n}}{|a_{\mathbf{n},\mathbf{i}}|}^p=O\left({|\mathbf{n}|}^{\delta }\right)\mathit{\text{for }}0<\delta <1.$$

(2.12)

If Theorem  2.2(iii) holds, then (2.2) holds.

Proof From (2.12), (2.8) follows. Hence, by Corollary 2.6, we obtain (2.2). □

Remark Linear random fields are of great importance in time series analysis. They arise in a wide variety of context. Applications to economics, engineering, and physical science are extremely broad (see Kim et al. [15]).

Let $Y_{\mathbf{k}}={\sum }_{\mathbf{i}\ge \mathbf{1}}{a}_{\mathbf{k}+\mathbf{i}}{X}_{\mathbf{i}}$, where $\{a_{\mathbf{i}},\mathbf{i}\in Z_{+}^d\}$ is a field of real numbers with ${\sum }_{\mathbf{i}}\ge \mathbf{1}|a_{\mathbf{i}}|<\mathrm{\infty }$, and $\{X_{\mathbf{i}},\mathbf{i}\in Z_{+}^d\}$ is a field of the martingale difference random variables.

Define $a_{\mathbf{n},\mathbf{i}}={\sum }_{\mathbf{1}\le \mathbf{k}\le \mathbf{n}}{a}_{\mathbf{i}+\mathbf{k}}$. Then we have

$$\sum _{\mathbf{1}\le \mathbf{k}\le \mathbf{n}}{Y}_{\mathbf{k}}=\sum _{\mathbf{1}\le \mathbf{k}\le \mathbf{n}}\sum _{\mathbf{i}\ge \mathbf{1}}{a}_{\mathbf{i}+\mathbf{k}}{X}_{\mathbf{i}}=\sum _{\mathbf{i}\ge \mathbf{1}}\sum _{\mathbf{1}\le \mathbf{k}\le \mathbf{n}}{a}_{\mathbf{i}+\mathbf{k}}{X}_{\mathbf{i}}=\sum _{\mathbf{i}\ge \mathbf{1}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}.$$

Hence, we can use the above results to investigate the complete convergence for linear random fields.