Fixed points with finite variance of a smoothing transformation.
type: Item
instance of: scholarly article
title: Fixed points with finite variance of a smoothing transformation.
zbMATH Open document ID: 1075.60503
DOI: 10.1016/S0304-4149(03)00075-9
author: Amke Caliebe
author: Uwe Rösler
published in: Stochastic Processes and their Applications
publication date: 2005-11-29T00:00:00Z
review text: The paper deals with fixed points of a transformation of distributions of real random variables with finite variance that maps a distribution \(\mu \) to the distribution of \(\sum _{j=1}^\infty T_jW_j\), where \(W_1,W_2,\dots \) are independent random variables with distribution \(\mu \), and \(T_1,T_2,\dots \) is a sequence of real random variables independent of \(W_1,W_2,\dots \). Within the quite rich literature about distributional fixed points, the paper belongs to the small number of works not requiring the coefficients \(T_1,T_2,\dots \) to be nonnegative, instead it only requires \(E\sum _{j=1}^\infty T_j^2<\infty \) together with some condition assuring \(\sum _{j=1}^\infty T_jW_j\) to converge in \(L_2\). It concentrates on solutions with zero expectation, assuming the convergence condition \(E\sum _{j=1}^\infty | T_j| ^\beta <\infty \) for some \(\beta >2\). For that case, results about the existence and uniqueness of the solution and about the Lebesgue density of its absolutely continuous part are proven, and some investigation of its moment is performed. Moreover, a representation of the characteristic function of the solution by means of fixed points of a distributional transformation with coefficients \(T_1^2,T_2^2,\dots \) is derived, establishing a connection to the more traditionally investigated transformations with nonnegative coefficients. The proofs of the new results rely mainly on the martingale theory and on previous results by \textit{Q. Liu} [Adv. Appl. Probab. 29, 353--373 (1997; Zbl 0901.60055) and ibid. 30, 85--112 (1998; Zbl 0909.60075)], and \textit{J. D. Biggins} [J. Appl. Probab. 14, 630--636 (1977; Zbl 0373.60090)]. The results about the existence and uniqueness are actually formulated in a more general setting, covering also solutions with nonzero expectation, for which the convergence condition \(\sum _{j=1}^\infty T_j\in L_2\) is assumed. However, the proofs for the case of nonzero expectation basically consist in recalling pr…
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 2233783
zbMATH Keywords: distributional fixed point equations
zbMATH Keywords: functional equations
zbMATH Keywords: branching random walks
zbMATH Keywords: weighted branching processes
zbMATH Keywords: infinite particle systems
zbMATH Keywords: convergence of triangular schemes
MaRDI profile type: Publication
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description: scientific article