Complete moment convergence of double-indexed randomly weighted sums of mixing sequences
- Jian Han^{1}Email author and
- Yu Xiang^{2}
https://doi.org/10.1186/s13660-016-1260-4
© The Author(s) 2016
Received: 6 August 2016
Accepted: 22 November 2016
Published: 29 November 2016
Abstract
In this paper, we study the complete moment convergence of the sums of ρ̃-mixing sequences which are double-indexed randomly weighted and stochastically dominated by a random variable X. Under the different moment conditions on X and weights, many complete moment convergence and complete convergence results are obtained. Moreover, some simulations are given for illustration.
Keywords
MSC
1 Introduction
Obviously, one has \(0\leq\tilde{\rho}(k+1)\leq \tilde{\rho}(k)\leq1\) and \(\tilde{\rho}(0)=1\).
Definition 1
A sequence of random variables \(\{X_{n},n\geq1\}\) is said to be a ρ̃-mixing sequence if there exists \(k\in\mathbb{N}\) such that \(\tilde{\rho}(k)<1\).
The concept of ρ̃-mixing random variables dates back to Stein [1]. Bradley [2] studied the properties of ρ̃-mixing random variables and obtained the central limit theorem. There are many examples of the structure of ρ̃-mixing random variables. Let \(\{e_{n},n\geq1\}\) be a sequence of independent identically distributed (\(i.i.d.\)) random variables with zero mean and finite variance. For \(n\geq1\), let \(X_{n}=\sum_{i=0}^{p}c_{i}e_{n-i}\), where p is a positive integer and \(c_{i}\) are constants, \(i=0,1,2,\ldots,p\). It is known that \(\{X_{n}\}\) is a moving average process with order p. It can be checked that \(\{X_{n}\}\) is a ρ̃-mixing process. Moreover, if \(\{X_{n}\}\) is a strictly stationary, finite-state, irreducible, and aperiodic Markov chain, then it is a ρ̃-mixing sequence (see Bradley [3]). There are many results for ρ̃-mixing sequences; see Peligrad and Gut [4] and Utev and Peligrad [5] for the moment inequalities, Sung [6] and Hu et al. [7] for the inverse moments, Yang et al. [8] for the nonlinear regression model, Wang et al. [9] for the Bahadur representation, etc.
On the one hand, since Hsu and Robbins [10] gave the concept of complete convergence, it has been an important basic tool to study the convergence in probability and statistics. Baum and Katz [11] extended the complete convergence of Hsu and Robbins [10], Chow [12] first investigated the complete moment convergence. Many authors extend the results of complete convergence from the independent case to the dependent cases. For the strong convergence, complete convergence and the applications for NOD sequences, we can refer to Sung [13, 14], Wu [15], Chen et al. [16] among others. Similarly, for NSD sequences, they are referenced by Shen et al. [17], Wang et al. [18], Deng et al. [19], Shen et al. [20], Wang et al. [21] among others. For END sequences, we can refer to Wang et al. [22], Hu et al. [23], Wang et al. [24], etc. For more results of strong convergence, complete convergence and the applications, one can refer to Hu et al. [25], Rosalsky and Volodin [26], Wang et al. [27], Wu et al. [28], Yang et al. [29], Yang and Hu [30], Wang et al. [31] and so on. In addition, for ρ̃-mixing sequences, we can refer to Kuczmaszewska [32], An and Yuan [33], Wang et al. [34], Sung [35], Wu et al. [36] for the study of convergence and applications.
On the other hand, there are many authors who study the convergence of random variables which are randomly weighted. For example, Thanh and Yin [37] established the almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces; Thanh et al. [38] investigated complete convergence of the randomly weighted sums of ρ̃-mixing sequences and gave the application to linear-time-invariant systems; Cabrera et al. [39] investigated the conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables; Shen et al. [40] obtained the conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability. For randomly weighted sums of martingales differences, Yang et al. [41] and Yao and Lin [42] obtained some results of complete convergence and the moment of the maximum normed sums. For the tail behavior and ruin theory of randomly weighted sums of random variables, we can refer to Gao and Wang et al. [43], Tang and Yuan [44], Leng and Hu [45], Yang et al. [46], Mao and Ng [47] and the references therein.
For \(n\geq1\), let \(S_{n}=\sum_{i=1}^{n}A_{ni}X_{i}\), where \(\{X_{i}\} \) is a ρ̃-mixing sequence and \(\{A_{ni}\}\) are the double-indexed randomly weights. Inspired by the papers above, we study the complete moment convergence of the sums \(S_{n}\) of ρ̃-mixing sequences \(\{X_{i}\}\) which are double-indexed randomly weighted and stochastically dominated by a random variable X. Under the different moment conditions on X and weights, many complete moment convergence results are obtained. Moreover, some simulations are given for illustration. For the details, please see our results and simulations in Section 3. Some lemmas are presented in Section 2. Finally, the proofs of the main results are presented in Section 4. For a given ρ̃-mixing sequence of random variables \(\{X_{n},n\geq1\}\), denote the dependence coefficient \(\tilde{\rho}(k)\) by \(\tilde{\rho}(X,k)\). In addition, for convenience, let \(C,C_{1},C_{2},\ldots\) be some positive constants, which are independent of n and may have different values in different expressions, \(x^{+}=\max{(x,0)}\) and \(x^{-}=\max(-x,0)\).
2 Some lemmas
Lemma 2.1
Utev and Peligrad [5]
Lemma 2.2
Thanh et al. [38]
Let \(0\leq r<1\) and k be a positive integer. Let \(X=\{X_{n}, n\geq1\}\) and \(Y=\{Y_{n},n\geq1\}\) be two sequences of ρ̃-mixing random variables satisfying \(\tilde{\rho}(X,k)\leq r\) and \(\tilde{\rho}(Y,k)\leq r\), respectively. Suppose that \(f:\mathbb{R}\times\mathbb {R}\rightarrow\mathbb{R}\) is a Borel function. Assume that X is independent of Y. Then the sequence \(f(X,Y)=\{f(X_{n},Y_{n}),n\geq1\}\) is also a ρ̃-mixing sequence of random variables satisfying \(\tilde{\rho}(f(X,Y),k)\leq r\).
Lemma 2.3
Sung [48]
Lemma 2.4
Adler and Rosalsky [49] and Adler et al. [50]
3 Main results and simulations
Theorem 3.1
Theorem 3.2
For the case \(1\leq l<2\), we take \(p=2l\) and \(\alpha=2/p\) in Theorem 3.2 and obtain following result.
Theorem 3.3
When \(\alpha\geq1\) and \(E|X|<\infty\), we have the following result.
Theorem 3.4
Remark 3.1
In Theorem 3.12 of Thanh et al. [38], the authors obtained the complete convergence results of (3.3) in Theorems 3.1 and 3.2, and (3.7) and (3.8) in Theorem 3.3. But we also obtain the complete moment convergence results of (3.2) in Theorems 3.1 and 3.2, and (3.6) in Theorem 3.3. Meanwhile, we have the complete convergence result (3.9) in Theorem 3.4 under the first moment condition on X, so we extend the results of Thanh et al. [38]. In addition, if \(A_{ni}\equiv1\), then Wang et al. [27] established (3.2) and (3.3) for φ-mixing sequences (see Corollaries 3.2 and 3.3 of Wang et al. [27]). Similarly, if \(A_{ni}=a_{ni}\) are constant weights in (3.4) and (3.5), then Yang et al. [29] obtained (3.2), (3.6), and (3.8) for martingale differences (see Theorem 5 and Corollary 6 of Yang et al. [29]). Therefore, we extend the results of Wang et al. [27] and Yang et al. [29] to ρ̃-mixing sequences which are double-indexed randomly weighted. Moreover, some simulations are presented to illustrate (3.8).
Simulation 3.1
(1) For all \(n\geq1\), let \(\{A_{ni},1\leq i\leq n\}\) be \(i.i.d\). random variables satisfying \(A_{11}\sim t(m)\) with \(m>0\), which are also independent of \(\{e_{i}\}\).
4 The proofs of main results
Proof of Theorem 3.1
Consequently, combining (4.2) with (4.4)-(4.8), we get (3.2) immediately. Moreover, by Remark 2.6 of Sung [48], (3.3) also holds true. □
Proof of Theorem 3.2
Consequently, by (4.10) and (4.11)-(4.14), we obtain \(I_{1}<\infty\). So we have (3.2). Similarly, combining Remark 2.6 of Sung [48] with (3.2), we obtain (3.3). □
Proof of Theorem 3.3
On the one hand, by \(p=2l\), \(\alpha=2/p\), we have \(\alpha p=2\). On the other hand, by the fact \(1\leq l<2\), we have that (3.4) is the same as (3.5). Then, as an application of Theorem 3.2, we obtain (3.6) immediately. Moreover, by (3.3) with \(\alpha p=2\), we establish (3.7). Finally, by the Borel-Cantelli lemma, (3.8) holds true. □
Proof of Theorem 3.4
Declarations
Acknowledgements
The authors are deeply grateful to editors and anonymous referee for their careful reading and insightful comments, which helped in improving the earlier version of this paper.
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
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