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Implicit Renewal Theory and Power Tails on Trees

Part of: Stochastic analysis Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  04 January 2016

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.
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: [email protected]
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Abstract

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We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C1, C2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C1, C2,…); here ‘∨’ denotes the maximum operator.

Keywords

Implicit renewal theoryweighted branching processmultiplicative cascadestochastic recursionpower lawlarge deviationsstochastic fixed-point equation

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 528 - 561
Copyright
© Applied Probability Trust 

Footnotes

Supported by the NSF, grant no. CMMI-1131053.

Supported by the NSF, grant no. CMMI-1131053.

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