The Annals of Applied Probability

Regular Variation in the Tail Behaviour of Solutions of Random Difference Equations

D. R. Grey

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 4, Number 1 (1994), 169-183.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005205

Digital Object Identifier
doi:10.1214/aoap/1177005205

Mathematical Reviews number (MathSciNet)
MR1258178

Zentralblatt MATH identifier
0802.60057

JSTOR
links.jstor.org

Subjects
Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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

Citation

Grey, D. R. Regular Variation in the Tail Behaviour of Solutions of Random Difference Equations. Ann. Appl. Probab. 4 (1994), no. 1, 169--183. doi:10.1214/aoap/1177005205. https://projecteuclid.org/euclid.aoap/1177005205


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