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Implicit Renewal Theory and Power Tails on Trees

Published online by Cambridge University Press:  04 January 2016

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: molvera@ieor.columbia.edu
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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.

Type
General Applied Probability
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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