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Convergence Rates in the Implicit Renewal Theorem on Trees

Part of: Markov processes Special processes Limit theorems Stochastic analysis

Published online by Cambridge University Press:  30 January 2018

Predrag R. Jelenković  and
Mariana Olvera-Cravioto
Predrag R. Jelenković*
Affiliation:
Columbia University
Mariana Olvera-Cravioto*
Affiliation:
Columbia University
*
∗ Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA. Email address: predrag@ee.columbia.edu
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu
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Abstract

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We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

Keywords

Implicit renewal theoryweighted branching processbranching random walkmultiplicative cascaderate of convergencesmoothing transformstochastic recursionpower lawlarge deviationstochastic fixed-point equation

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60F10: Large deviations 60K05: Renewal theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1077 - 1088
Copyright
© Applied Probability Trust 

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