Limit theorems for continuous-time random walks in the double-array limit scheme
type: Item
instance of: scholarly article
title: Limit theorems for continuous-time random walks in the double-array limit scheme
zbMATH Open document ID: 1118.60013
published in: Journal of Mathematical Sciences (New York)
publication date: 2007-04-19T00:00:00Z
review text: Let \(\{X_{n,j}\},\) \(n,j=1,2,\dots,\) be a double array of rowwise i.i.d. real-valued random variables, and \(\{T_{n,j}\}\), \(n,j=1,2,\dots \), a double array of i.i.d. nonnegative random variables. For \(t\geq 0\) consider the sequence of continuous-time random walks defined by \(S_{n}( t) =\sum_{j=1}^{N_{n}( t) }X_{n,j}\), where \(N_{n}( t) \) is a renewal process defined by the relation \(\sum_{j=1}^{N_{n}( t) }T_{n,j}\leq t<\sum_{j=1}^{N_{n}( t) +1}T_{n,j}\), \(t\geq 0,\) for any \(n\geq 1\). Continuing work on continuous-time random walk, mainly done by them, the authors extend asymptotic results obtained in the classical case of degenerate \(X\)- and \(T\)-arrays, that is, where both \(X_{n,j}\) and \(T_{n,j}\) do not depend on \(n\). See, e.g., \textit{V. Kolokoltsov, V. Korolev} and \textit{V. Uchaikin} [J. Math. Sci., New York 105, No. 6, 2569--2576 (2001; Zbl 0994.60010)] for previous work. An important auxiliary result of the paper consists of two general theorems on the stability of the representation of a distribution function as a special mixture of stable laws.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 5144810
zbMATH Keywords: random walk
zbMATH Keywords: double-array
zbMATH Keywords: transfer theorem
zbMATH Keywords: stability
author: Vladimir E. Bening
author: V. Yu. Korolev
author: Vassili Kolokoltsov
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: s10958-006-0300-x
OpenAlex ID: W4253608872
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DOI: 10.1007/S10958-006-0300-X
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description: scientific article; zbMATH DE number 5144810