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Implicit renewal theory in the arithmetic case

Part of: Special processes Stochastic analysis

Published online by Cambridge University Press:  15 September 2017

Péter Kevei
Péter Kevei*
Affiliation:
Technische Universität München and MTA–SZTE Analysis and Stochastics Research Group
*
* Postal address: University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary. Email address: [email protected]
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Abstract

We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution X of the fixed point equations X =DAX + B and X =DAX ∨ B is ℓ(x) q(x) x-κ, where q is a logarithmically periodic function q(x eh) = q(x), x > 0, with h being the span of the arithmetic distribution of log A, and ℓ is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevičius (1975) and Goldie (1991).

Keywords

Perpetuity equationmaximum of perturbed random walkimplicit renewal theoremarithmetic distributioniterated function system

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 732 - 749
Copyright
Copyright © Applied Probability Trust 2017 

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