Open Access
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 Ψ(R) for random but nonlinear Ψ and where M may be negative.

Citation

Download Citation

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

Vol.4 • No. 1 • February, 1994
Back to Top