## The Annals of Applied Probability

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

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

#### 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