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Fixed points of a generalized smoothing transformation and applications to the branching random walk

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Quansheng Liu
Quansheng Liu*
Affiliation:
Université de Rennes 1
*
∗ Postal address: Institut Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France.
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Abstract

Let {Ai : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑i=1∞AiZi, where the Zi are independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points δ0 and δ∞ is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

Keywords

Smoothing transformationbranching processesbranching random walksMandelbrot's martingalefunctional equationstable law

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 85 - 112
Copyright
Copyright © Applied Probability Trust 1998 

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