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Regular variation in the tail behaviour of solutions of random difference equations. (English) Zbl 0802.60057
Author’s summary: 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.
Reviewer: J.H.Kim (Pusan)

##### MSC:
 60H25 Random operators and equations (aspects of stochastic analysis) 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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