On a theorem of Breiman and a class of random difference equations.(English)Zbl 1141.60041

Consider the tail behavior of the product of two independent nonnegative random variables $$X$$ and $$Y$$. Assume that $$X$$ is a regularly varying random variable with index $$\alpha$$. The tail behavior of the product $$X Y$$ is investigated when $$Y$$ satisfies conditions which are weaker than $$E [Y^{\alpha+\varepsilon}] < +\infty$$ for some $$\varepsilon > 0$$ (called Breiman condition). This work continues studies of [L. Breiman, Theor. Probab. Appl. 10, 323–331 (1965; Zbl 0147.37004)] and also applies its results to random difference equations.
It would be interesting to extend the studies to the case of dependent random variables $$X$$ and $$Y$$.

MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 60F10 Large deviations

Zbl 0147.37004
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