STOCHASTIC PROCESSES AND THEIR APPLICATIONS | 卷:130 |
Regular variation of fixed points of the smoothing transform | |
Article | |
Liang, Xingang1  Liu, Quansheng2,3  | |
[1] Beijing Technol & Business Univ, Sch Sci, Beijing 100048, Peoples R China | |
[2] Univ Bretagne Sud, LMBA, UMR CNRS 6205, F-56000 Vannes, France | |
[3] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Peoples R China | |
关键词: Tail behavior; Regular variation; Smoothing transform; Branching random walk; Mandelbrot's martingale; | |
DOI : 10.1016/j.spa.2019.11.011 | |
来源: Elsevier | |
【 摘 要 】
Let (N, A(1), A(2), ...) be a sequence of random variables with N is an element of N U {infinity} and A(i) is an element of R+. We are interested in asymptotic properties of non-negative solutions of the distributional equation Z =((d)) Sigma(N)(i=1) A(i)Z(i), where Z(i) are non-negative random variables independent of each other and independent of (N, A(1), A(2), ...), each having the same distribution as Z which is unknown. For a solution Z with finite mean, we prove that for a given alpha > 1, P(Z > x) is a function regularly varying at infinity of index -alpha if and only if the same is true for P(Y-1 > x), where Y-1 = Sigma(N)(i=1) A(i). The result completes the sufficient condition obtained by Iksanov & Polotskiy (2006) on the branching random walk. A similar result on sufficient condition is also established for the case where alpha =1. (C) 2019 Elsevier B.V. All rights reserved.
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