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On a Theorem of Breiman and a Class of Random Difference Equations

Part of: Limit theorems Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Denis Denisov  and
Bert Zwart
Denis Denisov*
Affiliation:
EURANDOM
Bert Zwart*
Affiliation:
Georgia Institute of Technology
*
∗Current address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
∗∗Postal address: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA 30332-0205, USA.
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Abstract

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We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.

Keywords

Regular variationsubexponential distributionrandom difference equation

MSC classification

Secondary: 60H25: Random operators and equations 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1031 - 1046
Copyright
Copyright © Applied Probability Trust 2007 

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