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Harmonic moments of inhomogeneous branching processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Didier Piau
Didier Piau*
Affiliation:
Université Lyon 1
*
∗ Current address: Institut Fourier UMR 5582, Université Joseph Fourier Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint Martin d'Hères, France. Email address: didier.piau@ujf-grenoble.fr
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Abstract

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We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k>A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

Keywords

Branching processharmonic momentinhomogeneous Markov chainpolymerase chain reaction

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 465 - 486
Copyright
Copyright © Applied Probability Trust 2006 

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