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

Wang, Dingcheng; Tang, Qihe

Stoch. Models 22, No. 2, 253-272 (2006).

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.

Reviewer: Ryszard Doman (Poznan)

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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