Regular variation of fixed points of the smoothing transform

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.