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February, 1994 Regular Variation in the Tail Behaviour of Solutions of Random Difference Equations
D. R. Grey
Ann. Appl. Probab. 4(1): 169-183 (February, 1994). DOI: 10.1214/aoap/1177005205

Abstract

Let $Q$ and $M$ be random variables with given joint distribution. Under some conditions on this joint distribution, there will be exactly one distribution for another random variable $R$, independent of $(Q,M)$, with the property that $Q + MR$ has the same distribution as $R$. When $M$ is nonnegative and satisfies some moment conditions, we give an improved proof that if the upper tail of the distribution of $Q$ is regularly varying, then the upper tail of the distribution of $R$ behaves similarly; this proof also yields a converse. We also give an application to random environment branching processes, and consider extensions to cases where $Q + MR$ is replaced by $\Psi(R)$ for random but nonlinear $\Psi$ and where $M$ may be negative.

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D. R. Grey. "Regular Variation in the Tail Behaviour of Solutions of Random Difference Equations." Ann. Appl. Probab. 4 (1) 169 - 183, February, 1994. https://doi.org/10.1214/aoap/1177005205

Information

Published: February, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0802.60057
MathSciNet: MR1258178
Digital Object Identifier: 10.1214/aoap/1177005205

Subjects:
Primary: 60H25
Secondary: 60J80

Keywords: random environment branching processes , Random equations , random recurrence relations , regular variation

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 1 • February, 1994
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