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Asymptotics of random contractions. (English) Zbl 1231.91196
Summary: We discuss the asymptotic behaviour of random contractions $$X=RS$$, where $$R$$, with distribution function $$F$$, is a positive random variable independent of $$S\in (0,1)$$. Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of $$X$$ assuming that $$F$$ is in the max-domain of attraction of an extreme value distribution and the distribution function of $$S$$ satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60F05 Central limit and other weak theorems 60G70 Extreme value theory; extremal stochastic processes 91B25 Asset pricing models (MSC2010)
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