## The Annals of Applied Probability

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

D. R. Grey

#### 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