## Tail probabilities of randomly weighted sums of random variables with dominated variation.(English)Zbl 1095.60008

The authors investigate the asymptotic relations $\text{Pr}\left(\sum\limits^{n}_{i=1}{\theta}_{i}X_{i}>x\right)\sim \sum\limits^{n}_{i=1}\text{Pr}({\theta}_{i}X_{i}>x)\sim \text{Pr}\left(\max\limits_{1\leq k\leq n}\sum\limits^{k}_{i=1}{\theta}_{i}X_{i}>x\right),$ as $$x\to \infty$$. Here $$X_{i}$$, $$i= 1,2,\dots$$, are independent, identically distributed real-valued random variables, $$\theta_{i}$$, $$i= 1,2,\dots$$, are nonnegative random variables independent of the sequence $$\{X_{i}; i= 1,2,\dots\}$$, and no specific assumptions on the dependence structure of the sequence $$\{\theta_{i}; i= 1,2,\dots\}$$ are made. The relations are proven to hold under the assumption that common distribution of $$X_{i}$$ belongs to some class of heavily-tailed distributions, and $$\theta_{i}$$ satisfy some moment conditions. The obtained results are applied to derive asymptotic estimates for the finite and infinite time ruin probabilities in a discrete time risk model with (possibly dependent) stochastic return rates.

### MSC:

 60E05 Probability distributions: general theory 60G70 Extreme value theory; extremal stochastic processes 91B30 Risk theory, insurance (MSC2010)

### Keywords:

heavy tail; ruin probability
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### References:

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