On the existence of \(\varphi\)-moments of the limit of a normalized supercritical Galton-Watson process
type: Item
instance of: scholarly article
title: On the existence of \(\varphi\)-moments of the limit of a normalized supercritical Galton-Watson process
zbMATH Open document ID: 1071.60075
author: Gerold Alsmeyer
author: Uwe Rösler
published in: Journal of Theoretical Probability
publication date: 2005-04-07T00:00:00Z
review text: Let \((\mathbb{Z})_{n\geq 0}\) be a supercritical Galton-Watson process with offspring distribution \((p_j)_{j\geq 0}\) and finite mean offspring \(\mu\). The normalized process \(W_n\overset{\text{def}} = \mu^{-n}\mathbb{Z}_n\), \(n\geq 0\), is a nonnegative and thus a.s. convergent martingale with limit \(W\). The Kesten-Stigum theorem states that in order for \(W\) to be positive on the set of non-extinction it is necessary and sufficient that \(EZ_1\log Z_1= \sum_{j\geq 1}p_j\log j<\infty\). K. B. Athreya (1971) showed for any \(p\geq 0\) that \(0< EW|\log W|^p<\infty\) holds if and only if \(EZ_1\log Z_1< \infty\). Also, he proved this equivalence to be true for any integrable solution \(W\) of the stochastic equation \[ W\overset {\text{d}}={1\over \mu} \sum^{Z_1}_{k=1} W(k).\tag{1} \] This article extends these results by providing a necessary and sufficient moment condition on \((p_j)_{j\geq 0}\) for the existence of \(E\varphi(W)\) for an even larger class of functions \(\varphi\). Given any nondecreasing convex function \(\varphi:[0,\infty)\to [0,\infty)\), the authors define the operator \(\mathbb{L}\) through \[ \mathbb{L}\varphi(x)\overset{\text{def}} = \int^s_0\int^s_0 {\varphi'(r)\over r} dr\,ds,\quad x\geq 0.\tag{2} \] Theorem 1.1. Let \((\mathbb{Z}_n)_{n\geq 0}\) be a supercritical Galton-Watson process with offspring distribution \((p_j)_{j\geq 0}\); finite mean offspring \(\mu\) and normalized limit \(W= \lim_{n\to\infty} \mu^{-n}\mathbb{Z}_n\). Then for each \(\varphi\in C^*\) \[ 0< E\varphi(W)<\infty\quad\text{iff}\quad E(\varphi(Z_1))< \infty \] holds true with \(\mathbb{L}\varphi\) as in (2). Theorem 1.2. In the situation of Theorem 1.1 the following assertions are equivalent: \[ 0< E\varphi(W)<\infty;\quad \sup_{n\geq 0} E\varphi(W_n)< \infty, \] \((\varphi(W_n))_{n\geq 0}\) is uniformly integrable, \[ \lim_{n\to\infty} E|\varphi(W_n)- \varphi(W)|< \infty,\;E\varphi\Biggl(\sup_{n\geq 0} W_n\Biggr)< \infty. \] Section 2 contains a discussion o…
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 2153672
zbMATH Keywords: convex function
zbMATH Keywords: regular variation
zbMATH Keywords: martingale
zbMATH Keywords: convex function inequality
zbMATH Keywords: Kesten-Stigum theorem
MaRDI profile type: Publication
full work available at URL: s10959-004-0582-1
OpenAlex ID: W1999134532
cites work: On generalized renewal measures and certain first passage times
cites work: The best constant in the Topchii-Vatutin inequality for martingales.
cites work: Q3664178
cites work: A note on a functional equation arising in Galton-Watson branching processes
cites work: Q5674726
cites work: Asymptotic properties of supercritical branching processes I: The Galton-Watson process
cites work: Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes
cites work: Q4726487
cites work: Q4359973
cites work: Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes
cites work: Convergence conditions for weighted branching processes
DOI: 10.1007/S10959-004-0582-1
description: scientific article; zbMATH DE number 2153672