Regular variation of fixed points of the smoothing transform

Xingang Liang and Quansheng Liu

Stochastic Processes and their Applications, 2020, vol. 130, issue 7, 4104-4140

Abstract: Let (N,A1,A2,…) be a sequence of random variables with N∈N∪{∞} and Ai∈R+. We are interested in asymptotic properties of non-negative solutions of the distributional equation Z=(d)∑i=1NAiZi, where Zi are non-negative random variables independent of each other and independent of (N,A1,A2,…), each having the same distribution as Z which is unknown. For a solution Z with finite mean, we prove that for a given α>1, P(Z>x) is a function regularly varying at ∞ of index −α if and only if the same is true for P(Y1>x), where Y1=∑i=1NAi. 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 α=1.

Keywords: Tail behavior; Regular variation; Smoothing transform; Branching random walk; Mandelbrot’s martingale (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2019.11.011

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