Elementary fixed points of the BRW smoothing transforms with infinite number of summands
type: Item
instance of: scholarly article
title: Elementary fixed points of the BRW smoothing transforms with infinite number of summands
zbMATH Open document ID: 1081.60017
author: Alexander Iksanov
published in: Stochastic Processes and their Applications
publication date: 2005-08-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0310244
review text: The branching random walk smoothing transform \(\mathbb{T}\) is defined as \[ \mathbb{T}: \text{distr}(U_1)\to\text{distr}\left(\sum^L_{i=1}X_iU_i\right), \] where given realizations \(\{X_i\}^L_{i=1}\) of a point process, \(U_1, U_2,\dots\), are conditionally independent identically distributed random variables, and \(0\leq\text{Prob} \{L=\infty\}\leq 1\). Necessary conditions for the existence of any nonnegative fixed points of \(\mathbb{T}\) are obtained. Moreover, there is a uniqueness result which proves that if an elementary fixed point exists, it is unique up to the scale. The notation of an elementary fixed point is introduced by some restrictions on its Laplace-Stieltjes transform. The tail behaviour of some fixed points with finite mean is studied.
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 2192536
zbMATH Keywords: branching random walk
reviewed by: V. V. Goryainov
MaRDI profile type: Publication
OpenAlex ID: W1971245385
arXiv ID: math/0310244
cites work: On the Supercritical One Dimensional Age Dependent Branching Processes
cites work: On a class of characterization problems for random convex combinations
cites work: Asymptotic laws for nonconservative self-similar fragmentations
cites work: Martingale convergence in the branching random walk
cites work: Q4718073
cites work: Seneta-Heyde norming in the branching random walk
cites work: Fixed points of the smoothing transform: the boundary case
cites work: Measure change in multitype branching
cites work: Q3992980
cites work: Symmetric fixed points of a smoothing transformation
cites work: Fixed points with finite variance of a smoothing transformation.
cites work: Fixed points of the smoothing transformation
cites work: Implicit renewal theory and tails of solutions of random equations
cites work: Stability of perpetuities
cites work: One limit distribution for a random walk on the line
cites work: Sur une extension de la notion de loi semi-stable. (On an extension of the notion of semi-stable law)
cites work: Thick and thin points for random recursive fractals
cites work: Random fractals and probability metrics
cites work: On positive distributions of the class \(L\) of self-decompsable laws
cites work: Q4431565
cites work: On fixed points of Poisson shot noise transforms
cites work: On a Pitman-Yor problem
cites work: New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms
cites work: Random difference equations and renewal theory for products of random matrices
cites work: Two renewal theorems for general random walks tending to infinity
cites work: Sur Une Équation Fonctionnelle Et SES Applications: Une Extension Du Théorème De Kesten-Stigum Concernant Des Processus De Branchement
cites work: Fixed points of a generalized smoothing transformation and applications to the branching random walk
cites work: On generalized multiplicative cascades
cites work: Q5284193
cites work: Probability on trees and networks
cites work: Random Recursive Constructions: Asymptotic Geometric and Topological Properties
cites work: Infinitely Divisible Laws Associated with Hyperbolic Functions
cites work: Probability metrics and recursive algorithms
cites work: Q4002350
cites work: A fixed point theorem for distributions
cites work: Q4937701
DOI: 10.1016/J.SPA.2004.06.002
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description: scientific article