Open Access

The moment of maximum normed randomly weighted sums of martingale differences

Journal of Inequalities and Applications20152015:264

https://doi.org/10.1186/s13660-015-0786-1

Received: 12 April 2015

Accepted: 13 August 2015

Published: 28 August 2015

Abstract

By using some inequalities and properties of martingale differences, we investigate the moment of maximum normed randomly weighted sums of martingale differences under some weakly conditions. A sufficient condition to the moment of this stochastic sequence with maximum norm is presented in this paper.

Keywords

random weightedmaximum normedmartingale differences

MSC

60F1560F25

1 Introduction

Let \(\{X_{n},n\geq1\}\) be a sequence, independent and identically distributed with \(EX_{1}=0\). Denote \(S_{n}=\sum_{i=1}^{n} X_{i}\), \(n\geq1\). For \(0< r<2\) and \(p>0\), it is well known that
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} E|X_{1}|^{r}< \infty,\quad \text{if } p< r, \\ E[|X_{1}|^{r}\log(1+|X_{1}|)]< \infty,\quad \text{if } p=r, \\ E|X_{1}|^{p}< \infty,\quad \text{if } p>r, \end{array}\displaystyle \right . \end{aligned}$$
(1.1)
$$\begin{aligned}& E \biggl(\sup_{n\geq1}\biggl\vert \frac{X_{n}}{n^{1/r}}\biggr\vert ^{p} \biggr)< \infty, \end{aligned}$$
(1.2)
$$\begin{aligned}& E \biggl(\sup_{n\geq1}\biggl\vert \frac{S_{n}}{n^{1/r}}\biggr\vert ^{p} \biggr)< \infty \end{aligned}$$
(1.3)
are all equivalent. Marcinkiewicz and Zygmund [1] obtain (1.1) (1.3) for the case \(p\geq r=1\), Burkerholder [2] gets (1.3) (1.1) for the case \(p=r=1\), Gut [3] proves that (1.1)-(1.3) are equivalent in the case \(p\geq r\), and Choi and Sung [4] show that (1.1)-(1.3) are equivalent in the case \(p< r\). For \(0< r<2\) and \(p>0\), Chen and Gan [5] prove that (1.1)-(1.3) are equivalent under the dependent case such as ρ-mixing random variables.

By using the method of dominated by a nonnegative random variable, we investigate the randomly weighted sums of martingale differences under some weakly conditions. A sufficient condition to (1.2) and (1.3) is presented. To a certain extent, we generalize the result of Chen and Gan [5] for ρ-mixing random variables to the case of randomly weighted sums of martingale differences. For the details, please see our main result of Theorem 2.1 in Section 2.

Recall that the sequence \(\{X_{n},n\geq1\}\) is stochastically dominated by a nonnegative random variable X if
$$\sup_{n\geq1}P\bigl(\vert X_{n}\vert >t\bigr)\leq CP(X>t)\quad \text{for some positive constant } C \text{ and all } t\geq0 $$
(see Adler and Rosalsky [6] and Adler et al. [7]). A bound on tail probabilities for quadratic forms in independent random variables is seen by using the following condition. There exist \(C>0\) and \(\gamma>0\) such that for all \(n\geq1\) and all \(x\geq0\), we have \(P(|X_{n}|\geq x)\leq C\int_{x}^{\infty}e^{-\gamma t^{2}}\, dt\). For details, see Hanson and Wright [8] and Wright [9].

Meanwhile, the definition of martingale differences can be found in many books such as Stout [10], Hall and Heyde [11], Shiryaev [12], Gut [13], and so on. There are many results of martingale differences. For example, Ghosal and Chandra [14] gave the complete convergence of martingale arrays; Stoica [15, 16] investigated the Baum-Katz-Nagaev-type results for martingale differences; Wang et al. [17] also studied the complete moment convergence for martingale differences; Yang et al. [18] obtained the complete convergence for the moving average process of martingale differences; Yang et al. [19] investigated the complete moment convergence for randomly weighted sums of martingale differences, etc.

On the other hand, randomly weighted sums have been an attractive research topic in the literature of applied probability. For example, Thanh and Yin [20] studied the almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces; Thanh et al. [21] investigated the convergence analysis of double-indexed and randomly weighted sums of mixing processes and gave its application to state observers of linear-time-invariant systems; Kevei and Mason [22] and Hormann and Swan [23] studied the asymptotic properties of randomly weighted sums and self-normalized sums; Cabrera et al. [24] and Shen et al. [25] investigated the conditional convergence for randomly weighted sums; Gao and Wang [26] and Tang and Yuan [27] investigated the randomly weighted sums of random variables and have given an application to ruin theory and capital allocation; Chen [28] obtained some asymptotically results of randomly weighted sums of dependent random variables with dominated variation, and so on.

Throughout the paper, \(I(A)\) is the indicator function of set A and \(C, C_{1}, C_{2},\ldots\) denote some positive constants not depending on n. The following lemmas are our basic techniques to prove our results.

Lemma 1.1

(cf. Hall and Heyde (Theorem 2.11 in [11]))

If \(\{X_{i},\mathscr{F}_{i},1\leq i\leq n\}\) is a martingale difference and \(p>0\), then there exists a constant C depending only on p such that
$$E \Biggl(\max_{1\leq k\leq n}\Biggl\vert \sum _{i=1}^{k}X_{i}\Biggr\vert ^{p} \Biggr)\leq C \Biggl\{ E \Biggl(\sum_{i=1}^{n}E \bigl(X_{i}^{2}|\mathscr{F}_{i-1}\bigr) \Biggr)^{p/2} +E \Bigl(\max_{1\leq i\leq n}|X_{i}|^{p} \Bigr) \Biggr\} , \quad n\geq1. $$

Lemma 1.2

(cf. Adler and Rosalsky (Lemma 1 in [6]) and Adler et al. (Lemma 3 in [7]))

Let \(\{ X_{n},n\geq 1\}\) be a sequence of random variables, which is stochastically dominated by a nonnegative random variable X. Then, for any \(\alpha>0\) and \(b>0\), the following two statements hold:
  • \(E[|X_{n}|^{\alpha}I(|X_{n}|\leq b)]\leq C_{1}\{E[X^{\alpha}I(X\leq b)]+b^{\alpha}P(X>b)\}\),

  • \(E[|X_{n}|^{\alpha}I(|X_{n}|>b)]\leq C_{2}E[X^{\alpha}I(X>b)]\),

where \(C_{1}\) and \(C_{2}\) are positive constants not depending on n. Consequently, for all \(n\geq1\), one has \(E|X_{n}|^{\alpha}\leq C_{3}EX^{\alpha}\), where \(C_{3}\) is a positive constant not depending on n.

2 The main result and its proof

Theorem 2.1

Let \(0< r<2\), \(0< p<2\), and \(\{X_{n},\mathscr{F}_{n},n\geq1\}\) be a martingale difference sequence, which is stochastically dominated by a nonnegative random variable X such that
$$ \left \{ \textstyle\begin{array}{l} \textit{for } p< r, \quad \left \{ \textstyle\begin{array}{l} C_{1}EX< \infty,\quad \textit{if } 0< r< 1, \\ C_{2}E[X\log(1+X)]< \infty, \quad \textit{if } r=1, \\ C_{3}EX^{r}< \infty, \quad \textit{if } r>1, \end{array}\displaystyle \right . \\ \textit{for } p=r,\quad \left \{ \textstyle\begin{array}{l} C_{4}EX< \infty, \quad \textit{if } 0< r< 1, \\ C_{5}E[X\log^{2}(1+X)]< \infty,\quad \textit{if } r=1, \\ C_{6}E[X^{r}\log(1+X)]< \infty,\quad \textit{if } r>1, \end{array}\displaystyle \right . \\ \textit{for } p>r,\quad \left \{ \textstyle\begin{array}{l} C_{7}EX< \infty, \quad \textit{if } 0< p< 1, \\ C_{8}E[X\log(1+X)]< \infty,\quad \textit{if } p=1, \\ C_{9}EX^{p}< \infty, \quad \textit{if } p>1. \end{array}\displaystyle \right . \end{array}\displaystyle \right . $$
(2.1)
Assume that \(\{A_{n},n\geq1\}\) is an independent sequence of random variables, which is also independent of the sequence \(\{X_{n},n\geq1\}\). In addition, it is assumed that
$$ \sum_{i=1}^{n}EA_{i}^{2}= O(n). $$
(2.2)
Let \(S_{n}=\sum_{i=1}^{n} A_{i}X_{i}\), \(n\geq1\). Then one has the result (1.3), which implies the result (1.2).

Remark 2.1

In Theorem 2.1, \(\{A_{n}X_{n},\mathscr{F}_{n},n\geq 1\}\) may be not a martingale difference, since \(A_{n}\) is not required to be measurable with respect to \(\mathscr{F}_{n-1}\). We use the property of independence and the method of martingales to study the moments of maximum normed (1.2) and (1.3) and give a sufficient condition (2.1) for them.

Proof

It can be argued that
$$\begin{aligned} E\biggl(\sup_{n\geq1}\biggl\vert \frac{S_{n}}{n^{1/r}}\biggr\vert ^{p}\biggr) =&\int_{0}^{\infty}P\biggl(\sup _{n\geq1}\biggl\vert \frac{S_{n}}{n^{1/r}}\biggr\vert >t^{1/p} \biggr)\,dt \\ \leq& 2^{p/r}+\int_{2^{p/r}}^{\infty}\sum _{k=1}^{\infty}P\biggl(\max_{2^{k-1}\leq n< 2^{k}} \biggl\vert \frac{S_{n}}{n^{1/r}}\biggr\vert >t^{1/p}\biggr)\,dt \\ \leq& 2^{p/r}+\int_{2^{p/r}}^{\infty}\sum _{k=1}^{\infty}P\Bigl(\max_{1\leq n\leq2^{k}}| S_{n}|>2^{(k-1)/r}t^{1/p}\Bigr)\,dt \quad \bigl(\text{let } s=2^{(k-1)p/r}t\bigr) \\ =& 2^{p/r}+2^{p/r}\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}P\Bigl(\max_{1\leq n\leq2^{k}}|S_{n}|>s^{1/p} \Bigr)\,ds. \end{aligned}$$
(2.3)
Let \(\mathscr{G}_{0}=\{\emptyset,\Omega\}\), \(\mathscr{G}_{n}=\sigma(X_{1},\ldots,X_{n})\), \(n\geq1\), and \(X_{si}=X_{i}I(|X_{i}|\leq s^{1/p})\), where \(s^{1/p}>0\). It can be argued that
$$A_{i}X_{i}=A_{i}X_{i}I \bigl(|X_{i}|>s^{1/p}\bigr)+\bigl[A_{i}X_{si}-E(A_{i}X_{si}| \mathscr {G}_{i-1})\bigr]+E(A_{i}X_{si}| \mathscr{G}_{i-1}), \quad 1\leq i\leq n. $$
Then
$$\begin{aligned}& \sum_{k=1}^{\infty}2^{-kp/r}\int _{2^{kp/r}}^{\infty}P\Bigl(\max_{1\leq n\leq 2^{k}}|S_{n}|>s^{1/p} \Bigr)\,ds \\& \quad \leq \sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}P\Biggl(\max_{1\leq n\leq2^{k}} \Biggl\vert \sum_{i=1}^{n} A_{i}X_{i}I\bigl(|X_{i}|>s^{1/p}\bigr) \Biggr\vert >s^{1/p}/2\Biggr)\,ds \\& \qquad {} +\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}P\Biggl(\max_{1\leq n\leq2^{k}} \Biggl\vert \sum_{i=1}^{n} \bigl[A_{i}X_{si}-E(A_{i}X_{si}| \mathscr{G}_{i-1})\bigr]\Biggr\vert >s^{1/p}/4 \Biggr)\,ds \\& \qquad {} +\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}P\Biggl(\max_{1\leq n\leq2^{k}} \Biggl\vert \sum_{i=1}^{n} E(A_{i}X_{si}|\mathscr{G}_{i-1})\Biggr\vert >s^{1/p}/4\Biggr)\,ds \\& \quad =: I_{1}+I_{2}+I_{3}. \end{aligned}$$
(2.4)
Combining Hölder’s inequality with (2.2), we have
$$ \sum_{i=1}^{n} E|A_{i}|\leq \Biggl(\sum_{i=1}^{n} EA_{i}^{2} \Biggr)^{1/2}\Biggl(\sum_{i=1}^{n} 1 \Biggr)^{1-1/2}=O(n). $$
(2.5)
In view of \(\{A_{n},n\geq1\}\) is independent of the sequence \(\{X_{n},n\geq1\}\); by Markov’s inequality, (2.5), and Lemma 1.2, we get
$$\begin{aligned} I_{1} \leq&2\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}s^{-1/p} E\Biggl(\max _{1\leq n\leq2^{k}}\Biggl\vert \sum_{i=1}^{n} A_{i}X_{i}I\bigl(|X_{i}|>s^{1/p}\bigr) \Biggr\vert \Biggr)\,ds \\ \leq&2\sum_{k=1}^{\infty}2^{-kp/r}\int _{2^{kp/r}}^{\infty}s^{-1/p}\Biggl(\sum _{i=1}^{2^{k}}E|A_{i}|E|X_{i}|I \bigl(|X_{i}|>s^{1/p}\bigr) \Biggr)\,ds \\ \leq&C_{1}\sum_{k=1}^{\infty}2^{k-kp/r}\sum_{m=k}^{\infty}\int _{2^{mp/r}}^{2^{(m+1)p/r}} s^{-1/p}E\bigl[XI \bigl(X>s^{1/p}\bigr)\bigr]\,ds \\ \leq& C_{2}\sum_{k=1}^{\infty}2^{k-kp/r}\sum_{m=k}^{\infty}2^{mp/r-m/r}E \bigl[XI\bigl(X>2^{m/r}\bigr)\bigr] \\ =&C_{2}\sum_{m=1}^{\infty}2^{mp/r-m/r}E \bigl[XI\bigl(X>2^{m/r}\bigr)\bigr]\sum_{k=1}^{m} 2^{k-kp/r} \\ \leq&\left \{ \textstyle\begin{array}{l} C_{3}\sum_{m=1}^{\infty}2^{m-m/r}E[XI(X>2^{m/r})], \quad \text{if } p< r, \\ C_{4}\sum_{m=1}^{\infty}m2^{m-m/r}E[XI(X>2^{m/r})],\quad \text{if } p=r, \\ C_{5}\sum_{m=1}^{\infty}2^{mp/r-m/r}E[XI(X>2^{m/r})],\quad \text{if } p>r. \end{array}\displaystyle \right . \end{aligned}$$
(2.6)
For the case \(p< r\), if \(0< r<1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}2^{m-m/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< Y \leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m(1-1/r)} \\& \quad \leq C_{1}\sum _{k=1}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq2^{(k+1)/r}\bigr)\bigr]\leq C_{1}EX. \end{aligned}$$
If \(r=1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}E\bigl[XI \bigl(X>2^{m}\bigr)\bigr] = \sum_{m=1}^{\infty}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k}< X\leq 2^{k+1}\bigr)\bigr] \\& \quad =\sum_{k=1}^{\infty}E\bigl[XI\bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr]\sum _{m=1}^{k}1 = \sum_{k=1}^{\infty}k E\bigl[XI \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr] \\& \quad \leq C_{2}\sum _{k=1}^{\infty}E\bigl[X\log(1+X)I \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr] \\& \quad \leq C_{2}E\bigl[X\log(1+X)\bigr]. \end{aligned}$$
Otherwise, for \(r>1\), one has
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}2^{m-m/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m-m/r} \\& \quad \leq C_{1}\sum _{k=1}^{\infty}2^{k-k/r} E\bigl[XI \bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq C_{3}\sum_{k=1}^{\infty}E\bigl[X^{r}I\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{3}EX^{r}. \end{aligned}$$
Similarly, for the case \(p=r\), if \(0< r<1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m2^{m-m/r} \sum_{k=m}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} m2^{m(1-1/r)} \\& \quad \leq C_{4}\sum_{k=1}^{\infty}E\bigl[XI\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{4}EX. \end{aligned}$$
If \(r=1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k}< X \leq2^{k+1}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr]\sum _{m=1}^{k} m \leq C_{5}\sum _{k=1}^{\infty}k^{2}E\bigl[XI \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr] \\& \quad \leq C_{5}\sum_{k=1}^{\infty}E\bigl[X\log^{2} (1+X)I\bigl(2^{k}< X\leq2^{k+1} \bigr)\bigr] \leq C_{5}E\bigl[X\log^{2} (1+X)\bigr]. \end{aligned}$$
Otherwise, for \(r>1\), it follows that
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m2^{m-m/r} \sum_{k=m}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr]k\sum _{m=1}^{k} 2^{m-m/r} \\& \quad \leq C_{6}\sum_{k=1}^{\infty}k2^{k-k/r} E\bigl[XI\bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr) \bigr] \leq C_{7}E\bigl[X^{r}\log(1+X)\bigr]. \end{aligned}$$
On the other hand, for the case \(p>r\), if \(0< p<1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{mp/r-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}2^{m(p-1)/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m(p-1)/r} \\& \quad \leq C_{8}\sum _{k=1}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq 2^{(k+1)/r}\bigr)\bigr] \leq C_{8}EX. \end{aligned}$$
If \(p=1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{mp/r-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 1 \\& \quad =\sum_{k=1}^{\infty}kE\bigl[XI\bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq C_{9}EX\log(1+X). \end{aligned}$$
For \(p>1\), one has
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{mp/r-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}2^{m(p-1)/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m(p-1)/r} \\& \quad \leq C_{10}\sum_{k=1}^{\infty}2^{k(p-1)/r}E \bigl[XI\bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq C_{10}\sum_{k=1}^{\infty}E\bigl[X^{p}I\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{10}EX^{p}. \end{aligned}$$
Consequently, in view of (2.6), the conditions of Theorem 2.1, and the inequalities above, we have
$$\begin{aligned} I_{1} \leq&\left \{ \textstyle\begin{array}{l} C_{1}\sum_{m=1}^{\infty}2^{m-m/r}E[XI(X>2^{m/r})], \quad \text{if } p< r, \\ C_{2}\sum_{m=1}^{\infty}m2^{m-m/r}E[XI(X>2^{m/r})],\quad \text{if } p=r, \\ C_{3}\sum_{m=1}^{\infty}2^{mp/r-m/r}E[XI(X>2^{m/r})], \quad \text{if } p>r \end{array}\displaystyle \right . \\ \leq&\left \{ \textstyle\begin{array}{l} \text{for } p< r,\quad \left \{ \textstyle\begin{array}{l} C_{4}EX< \infty,\quad \text{if } 0< r< 1, \\ C_{5}E[X\log(1+X)]< \infty, \quad \text{if } r=1, \\ C_{6}EX^{r}< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p=r,\quad \left \{ \textstyle\begin{array}{l} C_{7}EX< \infty, \quad \text{if } 0< r< 1, \\ C_{8}E[X\log^{2}(1+X)]< \infty,\quad \text{if } r=1, \\ C_{9}E[X^{r}\log(1+X)]< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p>r,\quad \left \{ \textstyle\begin{array}{l} C_{10}EX< \infty,\quad \text{if } 0< p< 1, \\ C_{11}E[X\log(1+X)]< \infty, \quad \text{if } p=1, \\ C_{12}EX^{p}< \infty, \quad \text{if } p>1. \end{array}\displaystyle \right . \end{array}\displaystyle \right . \end{aligned}$$
(2.7)
It can be checked that for the fixed real numbers \(a_{1}, \ldots, a_{n}\),
$$\bigl\{ a_{i}X_{si}-E(a_{i}X_{si}| \mathscr{G}_{i-1}),\mathscr{G}_{i},1\leq i\leq n\bigr\} $$
is also a martingale difference. So one has by Lemma 1.1
$$\begin{aligned}& E \Biggl\{ \max_{1\leq n\leq2^{k}}\sum_{i=1}^{n} \bigl[A_{i} X_{si}-E(A_{i}X_{si}| \mathscr{G}_{i-1})\bigr] \Biggr\} ^{2} \\& \quad = E \Biggl\{ E \Biggl(\max_{1\leq n\leq2^{k}}\sum _{i=1}^{n}\bigl[a_{i} X_{si}-E(a_{i}X_{si}| \mathscr{G}_{i-1})\bigr] \Biggr)^{2} \Big|A_{1}=a_{1}, \ldots ,A_{n}=a_{n} \Biggr\} \\& \quad \leq C_{1}E \Biggl\{ \sum_{i=1}^{2^{k}}E \bigl(a_{i}^{2}X_{si}^{2}\bigr) \Big|A_{1}=a_{1},\ldots,A_{n}=a_{n} \Biggr\} \\& \quad = C_{1}\sum_{i=1}^{2^{k}}EA_{i}^{2}EX_{si}^{2}, \end{aligned}$$
(2.8)
by using the fact that \(\{A_{1},\ldots, A_{n}\}\) is independent of \(\{X_{s1}, \ldots, X_{sn}\}\). Consequently, by Markov’s inequality, (2.2), (2.8), and Lemma 1.2, one can check that
$$\begin{aligned} I_{2} \leq&C_{1}\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}s^{-2/p}E \Biggl\{ \max _{1\leq n\leq2^{k}}\sum_{i=1}^{n} \bigl[A_{i} X_{si}-E(A_{i}X_{si}| \mathscr{G}_{i-1})\bigr] \Biggr\} ^{2}\,ds \\ \leq&C_{2}\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}s^{-2/p} \Biggl(\sum _{i=1}^{2^{k}}EA_{i}^{2}EX_{si}^{2} \Biggr)\,ds \\ \leq&C_{3}\sum_{k=1}^{\infty}2^{-kp/r+k}\int_{2^{kp/r}}^{\infty}s^{-2/p}E \bigl[X^{2}I\bigl(X\leq s^{1/p}\bigr)\bigr]\,ds \\ &{}+C_{4}\sum_{k=1}^{\infty}2^{-kp/r+k} \int_{2^{kp/r}}^{\infty}P\bigl(X>s^{1/p}\bigr)\,ds \\ =:&C_{3}I_{21}+C_{4}I_{22}. \end{aligned}$$
(2.9)
For \(I_{21}\), it follows that
$$\begin{aligned} I_{21} =&\sum_{k=1}^{\infty}2^{-kp/r+k}\sum_{m=k}^{\infty}\int _{2^{mp/r}}^{2^{(m+1)p/r}} s^{-2/p}E\bigl[X^{2}I \bigl(X\leq s^{1/p}\bigr)\bigr]\,ds \\ \leq&\sum_{k=1}^{\infty}2^{-kp/r+k}\sum _{m=k}^{\infty}2^{mp/r-2m/r}E \bigl[X^{2}I\bigl(X\leq 2^{(m+1)/r}\bigr)\bigr] \\ =&\sum_{m=1}^{\infty}2^{m(p-2)/r}E \bigl[X^{2}I\bigl(X\leq 2^{(m+1)/r}\bigr)\bigr]\sum _{k=1}^{m} 2^{k(1-p/r)}. \end{aligned}$$
If \(p< r\), one has by \(r<2\)
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m(p-2)/r}E \bigl[X^{2}I\bigl(X\leq 2^{(m+1)/r}\bigr)\bigr]\sum _{k=1}^{m} 2^{k(1-p/r)} \\& \quad \leq C_{1}\sum_{m=1}^{\infty}2^{m(r-2)/r}E \bigl[X^{2}I\bigl(X\leq2^{(m+1)/r}\bigr)\bigr] \\& \quad = C_{1}\sum_{m=1}^{\infty}2^{m(r-2)/r}E \bigl[X^{2}I\bigl(X\leq2^{1/r}\bigr)\bigr] \\& \qquad {} +C_{1}\sum_{m=1}^{\infty}2^{m(r-2)/r}\sum_{i=1}^{m}E \bigl[X^{2}I\bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr] \\& \quad \leq C_{2}+C_{1}\sum_{i=1}^{\infty}E \bigl[X^{2}I\bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr]\sum _{m=i}^{\infty}2^{m(r-2)/r} \\& \quad \leq C_{2}+C_{3}2^{(2-r)/r}\sum _{i=1}^{\infty }2^{(i+1)(r-2)/r}E\bigl[X^{2}I \bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr] \\& \quad \leq C_{4}+C_{5}EX^{r}. \end{aligned}$$
For the case \(p=r\), by \(r<2\), one has
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m(p-2)/r}E \bigl[X^{2}I\bigl(X\leq 2^{(m+1)/r}\bigr)\bigr]\sum _{k=1}^{m} 2^{k(1-p/r)} \\& \quad \leq C_{1}\sum_{m=1}^{\infty}m 2^{m(r-2)/r}E\bigl[X^{2}I\bigl(X\leq2^{(m+1)/r}\bigr)\bigr] \\& \quad = C_{1}\sum_{m=1}^{\infty}m2^{m(r-2)/r}E\bigl[X^{2}I\bigl(X\leq2^{1/r}\bigr)\bigr] \\& \qquad {} +C_{1}\sum_{m=1}^{\infty}m 2^{m(r-2)/r}\sum_{i=1}^{m}E \bigl[X^{2}I\bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr] \\& \quad \leq C_{2}+C_{1}\sum_{i=1}^{\infty}E \bigl[X^{2}I\bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr]\sum _{m=i}^{\infty}m 2^{m(r-2)/r} \\& \quad \leq C_{2}+C_{3}2^{(2-r)/r}\sum _{i=1}^{\infty }i2^{(i+1)(r-2)/r}E\bigl[X^{2}I \bigl(2^{i/r}< X\leq2^{(i+1)/r}\bigr)\bigr] \\& \quad \leq C_{4}+C_{5}E\bigl[X^{r}\log(1+X) \bigr]. \end{aligned}$$
For the case \(p>r\), it can be checked by \(p<2\) that
$$\sum_{m=1}^{\infty}2^{m(p-2)/r}E \bigl[X^{2}I\bigl(Y\leq 2^{(m+1)/r}\bigr)\bigr]\leq CEX^{p}. $$
Consequently, it follows that
$$\begin{aligned} I_{21} \leq&\left \{ \textstyle\begin{array}{l} C_{1}\sum_{m=1}^{\infty}2^{m(r-2)/r}E[X^{2}I(X\leq2^{(m+1)/r})], \quad \text{if } p< r, \\ C_{2}\sum_{m=1}^{\infty}m2^{m(r-2)/r}E[X^{2}I(X\leq2^{(m+1)/r})],\quad \text{if } p=r, \\ C_{3}\sum_{m=1}^{\infty}2^{m(p-2)/r}E[X^{2}I(Y\leq 2^{(m+1)/r})],\quad \text{if } p>r \end{array}\displaystyle \right . \\ \leq&\left \{ \textstyle\begin{array}{l} C_{4}+C_{5}EX^{r}< \infty,\quad \text{if } p< r, \\ C_{6}+C_{7}E[X^{r}\log(1+X)]< \infty, \quad \text{if } p=r, \\ C_{8}EX^{p}< \infty,\quad \text{if } p>r. \end{array}\displaystyle \right . \end{aligned}$$
(2.10)
On the other hand, similar to the proofs of (2.6) and (2.7), we obtain
$$\begin{aligned} I_{22} \leq& \sum_{k=1}^{\infty}2^{k-kp/r}\int_{2^{kp/r}}^{\infty}s^{-1/p}E \bigl[XI\bigl(X>s^{1/p}\bigr)\bigr]\,ds \\ \leq&C_{2}\sum_{m=1}^{\infty}2^{mp/r-m/r}E \bigl[XI\bigl(X>2^{m/r}\bigr)\bigr]\sum_{k=1}^{m} 2^{k-kp/r} \\ \leq&\left \{ \textstyle\begin{array}{l} C_{3}\sum_{m=1}^{\infty}2^{m-m/r}E[XI(X>2^{m/r})], \quad \text{if } p< r, \\ C_{4}\sum_{m=1}^{\infty}m2^{m-m/r}E[XI(X>2^{m/r})], \quad \text{if } p=r, \\ C_{5}\sum_{m=1}^{\infty}2^{mp/r-m/r}E[XI(X>2^{m/r})], \quad \text{if } p>r. \end{array}\displaystyle \right . \end{aligned}$$
(2.11)
For the case \(p< r\), if \(0< r<1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad = \sum_{m=1}^{\infty}2^{m-m/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m(1-1/r)} \\& \quad \leq C_{1}\sum_{k=1}^{\infty}E\bigl[XI\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{1}EX. \end{aligned}$$
If \(r=1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}E\bigl[XI \bigl(X>2^{m}\bigr)\bigr] = \sum_{m=1}^{\infty}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k}< X\leq 2^{k+1}\bigr)\bigr] \\& \quad =\sum_{k=1}^{\infty}E\bigl[XI\bigl(2^{k}< X\leq 2^{k+1}\bigr)\bigr]\sum _{m=1}^{k}1 \leq \sum_{k=1}^{\infty}k E\bigl[XI \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr] \\& \quad \leq C_{2}\sum _{k=1}^{\infty}E\bigl[X\log(1+X)I \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr] \leq C_{2}E\bigl[X\log(1+X)\bigr]. \end{aligned}$$
Otherwise, for \(r>1\), we have
$$\begin{aligned}& \sum_{m=1}^{\infty}2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad = \sum_{m=1}^{\infty}2^{m-m/r}\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k/r}< X \leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} 2^{m-m/r} \\& \quad \leq C_{3}\sum_{k=1}^{\infty}2^{k-k/r} E\bigl[XI\bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq C_{3}\sum_{k=1}^{\infty}E\bigl[X^{r}I\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{3}EX^{r}. \end{aligned}$$
Similarly, for the case \(p=r\), if \(0< r<1\), then
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m2^{m-m/r} \sum_{k=m}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} m2^{m(1-1/r)} \\& \quad \leq C_{4}\sum_{k=1}^{\infty}E\bigl[XI\bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \leq C_{4}EX. \end{aligned}$$
For \(r=1\), it follows that
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m\sum _{k=m}^{\infty}E\bigl[XI\bigl(2^{k}< X \leq2^{k+1}\bigr)\bigr] \\& \quad = \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k}< X\leq2^{k+1}\bigr)\bigr]\sum _{m=1}^{k} m \leq C_{5}\sum _{k=1}^{\infty}k^{2}E\bigl[XI \bigl(2^{k}< X\leq 2^{k+1}\bigr)\bigr] \\& \quad \leq C_{5}\sum_{k=1}^{\infty}E\bigl[X\log^{2} (1+X)I\bigl(2^{k}< X\leq2^{k+1} \bigr)\bigr] \leq C_{5}E\bigl[X\log^{2} (1+X)\bigr]. \end{aligned}$$
Otherwise, for \(r>1\), it follows that
$$\begin{aligned}& \sum_{m=1}^{\infty}m2^{m-m/r}E\bigl[XI \bigl(X>2^{m/r}\bigr)\bigr] \\& \quad =\sum_{m=1}^{\infty}m2^{m-m/r} \sum_{k=m}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq2^{(k+1)/r}\bigr)\bigr] \\& \quad \leq \sum_{k=1}^{\infty}E\bigl[XI \bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr)\bigr]\sum _{m=1}^{k} m2^{m-m/r} \\& \quad \leq C_{6}\sum_{k=1}^{\infty}k 2^{k-k/r} E\bigl[XI\bigl(2^{k/r}< X\leq 2^{(k+1)/r}\bigr) \bigr] \leq C_{7}E\bigl[X^{r}\log(1+X)\bigr]. \end{aligned}$$
Therefore, by (2.7) for case \(p>r\), (2.11), and the inequalities above, we obtain
$$\begin{aligned} I_{22} \leq&\left \{ \textstyle\begin{array}{l} C_{1}\sum_{m=1}^{\infty}2^{m-m/r}E[XI(X>2^{m/r})],\quad \text{if } p< r, \\ C_{2}\sum_{m=1}^{\infty}m2^{m-m/r}E[XI(X>2^{m/r})],\quad \text{if } p=r, \\ C_{3}\sum_{m=1}^{\infty}2^{mp/r-m/r}E[XI(X>2^{m/r})], \quad \text{if } p>r \end{array}\displaystyle \right . \\ \leq&\left \{ \textstyle\begin{array}{l} \text{for } p< r, \quad \left \{ \textstyle\begin{array}{l} C_{4}EX< \infty, \quad \text{if } 0< r< 1, \\ C_{5}E[X\log(1+X)]< \infty, \quad \text{if } r=1, \\ C_{6}EX^{r}< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p=r,\quad \left \{ \textstyle\begin{array}{l} C_{7}EX< \infty,\quad \text{if } 0< r< 1, \\ C_{8}E[X\log^{2}(1+X)]< \infty,\quad \text{if } r=1, \\ C_{9}E[X^{r}\log(1+X)]< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p>r,\quad \left \{ \textstyle\begin{array}{l} C_{10}EX< \infty,\quad \text{if } 0< p< 1, \\ C_{11}E[X\log(1+X)]< \infty,\quad \text{if } p=1, \\ C_{12}EX^{p}< \infty, \quad \text{if } p>1. \end{array}\displaystyle \right . \end{array}\displaystyle \right . \end{aligned}$$
(2.12)
Obviously, it can be seen that \(\{X_{n},\mathscr{G}_{n},n\geq1\}\) is also a martingale difference, since \(\{X_{n},\mathscr{F}_{n},n\geq 1\}\) is a martingale difference. Combining with the fact that \(\{A_{n},n\geq1\}\) is independent of \(\{X_{n},n\geq1\}\), we have
$$\begin{aligned} E(A_{i}X_{i}|\mathscr{G}_{i-1}) =&E \bigl[E(A_{i}X_{i}|\mathscr{G}_{i})|\mathscr {G}_{i-1}\bigr]=E\bigl[X_{i}E(A_{i}| \mathscr{G}_{i})|\mathscr{G}_{i-1}\bigr] \\ =&EA_{i}E[X_{i}|\mathscr{G}_{i-1}]=0,\quad \mbox{a.s.}, 1\leq i\leq n. \end{aligned}$$
In view of the proofs of (2.6), (2.7), and the inequality above, we obtain by Markov’s inequality and (2.5)
$$\begin{aligned} I_{3} \leq&4\sum_{k=1}^{\infty}2^{-kp/r} \int_{2^{kp/r}}^{\infty}s^{-1/p} E\Biggl(\max _{1\leq n\leq2^{k}}\Biggl\vert \sum_{i=1}^{n} E(A_{i}X_{si}|\mathscr{G}_{i-1})\Biggr\vert \Biggr)\,ds \\ =&4\sum_{k=1}^{\infty}2^{-kp/r}\int _{2^{kp/r}}^{\infty}s^{-1/p} E\Biggl(\max _{1\leq n\leq2^{k}}\Biggl\vert \sum_{i=1}^{n} E\bigl(A_{i}X_{i}I\bigl(|X_{i}|\leq s^{1/p}\bigr)|\mathscr{G}_{i-1}\bigr)\Biggr\vert \Biggr)\,ds \\ =&4\sum_{k=1}^{\infty}2^{-kp/r}\int _{2^{kp/r}}^{\infty}s^{-1/p} E\Biggl(\max _{1\leq n\leq2^{k}}\Biggl\vert \sum_{i=1}^{n} E\bigl(A_{i}X_{i}I\bigl(|X_{i}|> s^{1/p}\bigr)|\mathscr{G}_{i-1}\bigr)\Biggr\vert \Biggr)\,ds \\ \leq&4\sum_{k=1}^{\infty}2^{-kp/r}\int _{2^{kp/r}}^{\infty}s^{-1/p} \sum _{i=1}^{2^{k}} E|A_{i}|E|X_{i}|I \bigl(|X_{i}|> s^{1/p}\bigr)\,ds \\ \leq&C_{1}\sum_{k=1}^{\infty}2^{k-kp/r} \int_{2^{kp/r}}^{\infty}s^{-1/p}E\bigl[XI \bigl(X>s^{1/p}\bigr)\bigr]\,ds \\ \leq&\left \{ \textstyle\begin{array}{l} \text{for } p< r, \quad \left \{ \textstyle\begin{array}{l} C_{1}EX< \infty,\quad \text{if } 0< r< 1, \\ C_{2}E[X\log(1+X)]< \infty,\quad \text{if } r=1, \\ C_{3}EX^{r}< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p=r,\quad \left \{ \textstyle\begin{array}{l} C_{4}EX< \infty, \quad \text{if } 0< r< 1, \\ C_{5}E[X\log^{2}(1+X)]< \infty,\quad \text{if } r=1, \\ C_{6}E[X^{r}\log(1+X)]< \infty, \quad \text{if } r>1, \end{array}\displaystyle \right . \\ \text{for } p>r, \quad \left \{ \textstyle\begin{array}{l} C_{7}EX< \infty, \quad \text{if } 0< p< 1, \\ C_{8}E[X\log(1+X)]< \infty,\quad \text{if } p=1, \\ C_{9}EX^{p}< \infty, \quad \text{if } p>1. \end{array}\displaystyle \right . \end{array}\displaystyle \right . \end{aligned}$$
(2.13)

Consequently, in view of (2.1), (2.3), (2.4), (2.7)-(2.10), (2.12), and (2.13), one has (1.3). By (1.3), it is easy to get (1.2). □

Declarations

Acknowledgements

The authors are deeply grateful to the editor and two anonymous referees, whose insightful comments and suggestions have contributed substantially to the improvement of this paper. This work is supported by the National Natural Science Foundation of China (11171188), National Social Science Fund of China (14ATJ005), the Humanities and Social Science Planning Foundation of Ministry of Education of China (14YJCZH155) and the Fundamental Scientific Research Funds for the Central Universities (2015HGZX0018).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Shandong University Qilu Securities Institute for Financial Studies, Shandong University
(2)
School of Mathematics, Hefei University of Technology
(3)
School of Mathematics, Shandong University

References

  1. Marcinkiewicz, J, Zygmund, A: Sur les fonctions independantes. Fundam. Math. 29, 60-90 (1937) Google Scholar
  2. Burkerholder, DL: Successive conditional expectations of an integrable function. Ann. Math. Stat. 33(3), 887-893 (1962) View ArticleGoogle Scholar
  3. Gut, A: Moments of the maximum of normed partial sums of random variables with multidimensional indices. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 205-220 (1979) MathSciNetView ArticleGoogle Scholar
  4. Choi, BD, Sung, SH: On moment conditions for the supremum of normed sums. Stoch. Process. Appl. 26, 99-106 (1987) MathSciNetView ArticleGoogle Scholar
  5. Chen, PY, Gan, SX: On moments of the maximum of normed partial sums of ρ-mixing random variables. Stat. Probab. Lett. 78(10), 1215-1221 (2008) MathSciNetView ArticleGoogle Scholar
  6. Adler, A, Rosalsky, A: Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl. 5(1), 1-16 (1987) MathSciNetView ArticleGoogle Scholar
  7. Adler, A, Rosalsky, A, Taylor, RL: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci. 12(3), 507-530 (1989) MathSciNetView ArticleGoogle Scholar
  8. Hanson, DL, Wright, FT: A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42(3), 1079-1083 (1971) MathSciNetView ArticleGoogle Scholar
  9. Wright, FT: A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1(6), 1068-1070 (1973) View ArticleGoogle Scholar
  10. Stout, WF: Almost Sure Convergence. Academic Press, New York (1974) MATHGoogle Scholar
  11. Hall, P, Heyde, CC: Martingale Limit Theory and Its Application. Academic Press, New York (1980) MATHGoogle Scholar
  12. Shiryaev, AN: Probability, 2nd edn. Springer, New York (1996) View ArticleGoogle Scholar
  13. Gut, A: Probability: A Graduate Course. Springer, Berlin (2005) Google Scholar
  14. Ghosal, S, Chandra, TK: Complete convergence of martingale arrays. J. Theor. Probab. 11(3), 621-631 (1998) MathSciNetView ArticleGoogle Scholar
  15. Stoica, G: Baum-Katz-Nagaev type results for martingales. J. Math. Anal. Appl. 336(2), 1489-1492 (2007) MathSciNetView ArticleGoogle Scholar
  16. Stoica, G: A note on the rate of convergence in the strong law of large numbers for martingales. J. Math. Anal. Appl. 381(2), 910-913 (2011) MathSciNetView ArticleMATHGoogle Scholar
  17. Wang, XJ, Hu, SH, Yang, WZ, Wang, XH: Convergence rates in the strong law of large numbers for martingale difference sequences. Abstr. Appl. Anal. 2012, Article ID 572493 (2012) MathSciNetGoogle Scholar
  18. Yang, WZ, Hu, SH, Wang, XJ: Complete convergence for moving average process of martingale differences. Discrete Dyn. Nat. Soc. 2012, Article ID 128492 (2012) MathSciNetGoogle Scholar
  19. Yang, WZ, Wang, YW, Wang, XH, Hu, SH: Complete moment convergence for randomly weighted sums of martingale differences. J. Inequal. Appl. 2013, 396 (2013) MathSciNetView ArticleGoogle Scholar
  20. Thanh, LV, Yin, G: Almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces. Taiwan. J. Math. 15(4), 1759-1781 (2011) MATHGoogle Scholar
  21. Thanh, LV, Yin, G, Wang, LY: State observers with random sampling times and convergence analysis of double-indexed and randomly weighted sums of mixing processes. SIAM J. Control Optim. 49(1), 106-124 (2011) MathSciNetView ArticleGoogle Scholar
  22. Kevei, P, Mason, DM: The asymptotic distribution of randomly weighted sums and self-normalized sums. Electron. J. Probab. 17, 46 (2012) MathSciNetGoogle Scholar
  23. Hormann, S, Swan, Y: A note on the normal approximation error for randomly weighted self-normalized sums. Period. Math. Hung. 67(2), 143-154 (2013) MathSciNetView ArticleGoogle Scholar
  24. Cabrera, MO, Rosalsky, A, Volodin, A: Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables. Test 21(2), 369-385 (2012) MathSciNetView ArticleMATHGoogle Scholar
  25. Shen, AT, Wu, RC, Chen, Y, Zhou, Y: Conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability. J. Inequal. Appl. 2013, 122 (2013) MathSciNetView ArticleGoogle Scholar
  26. Gao, QW, Wang, YB: Randomly weighted sums with dominated varying-tailed increments and application to risk theory. J. Korean Stat. Soc. 39(3), 305-314 (2010) View ArticleGoogle Scholar
  27. Tang, QH, Yuan, ZY: Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 17(3), 467-493 (2014) MathSciNetView ArticleGoogle Scholar
  28. Chen, DY: Randomly weighted sums of dependent random variables with dominated variation. J. Math. Anal. Appl. 420(2), 1617-1633 (2014) MathSciNetView ArticleGoogle Scholar

Copyright

© Yao and Lin 2015