On the rates of convergence in weak limit theorems for geometric random sums of the strictly stationary sequence of m -d
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Lithuanian Mathematical Journal
On the rates of convergence in weak limit theorems for geometric random sums of the strictly stationary sequence of m-dependent random variables∗ Tran Loc Hung a and Phan Tri Kien a,b a b
University of Finance and Marketing, 2/4 Tran Xuan Soan Street, District No. 7, Ho Chi Minh City, Vietnam University of Science – VNU. HCMC, 227 Nguyen Van Cu Street, District No. 5, Ho Chi Minh City, Vietnam (e-mail: [email protected])
Received January 25, 2019; revised May 24, 2019
Abstract. In this paper, we consider a strictly stationary sequence of m-dependent random variables through a compatible sequence of independent and identically distributed random variables by the moving averages processes. Using the Zolotarev distance, we estimate some rates of convergence in the weak limit theorems for normalized geometric random sums of the strictly stationary sequence of m-dependent random variables. The obtained results are extensions and generalizations of several known results on geometric random sums of independent and identically distributed random variables. MSC: geometric random sums, strictly stationary sequence, m-dependent random variables, Zolotarev’s distance Keywords: 60F05; 60G50; 60F99
1 Introduction The geometric random sums of independent and identically distributed (i.i.d.) random variables have a lot of applications in many various areas such as risk processes, ruin probability, queuing theory, reliability models, and so on. This problem has attracted much attention of many mathematicians like Klebanov (1984, 2003), Kruglov and Korolev (1990), Gnedenko and Kruglov (1996), Kalashnikov (1997), Sandhya and Pillai (2003), Kotz et al. (2001), Daly (2016), and so on (see [4, 6, 17, 18, 19, 20, 21, 28, 29]). The case of independent and nonidentically distributed have been considered by Toda (2012) in [31]. The mathematical tools have been used in research literature including method of inequalities, method of characteristic function, method of linear operators, Stein’s method and method of probability metrics, and so on (see [3, 13, 17, 18, 19, 20, 21, 22, 24, 28, 29]). It is worth pointing out that, up to the present, the obtained results have only just related to the cases of geometric random sums of i.i.d. random variables. ∗
The first author was supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 101.01-2010.02.
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T.L. Hung and P.T. Kien
An interesting question arises as to what happens if we have to deal with problems related to geometric random sums of m-dependent random variables from a strictly stationary sequence. An example of a strictly stationary sequence of m-dependent random variables can be given by a moving-average process (see, e.g., [30] and [15]). It is worth noting that when m = 0, a sequence of m-dependent random variables becomes a sequence of independent random variables; in particular, strictly stationary random variab
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