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.


60E05 Probability distributions: general theory
60G70 Extreme value theory; extremal stochastic processes
91B30 Risk theory, insurance (MSC2010)
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[1] Bingham N.H., Regular Variation (1987)
[2] DOI: 10.1137/1110037
[3] DOI: 10.1239/jap/1025131428 · Zbl 1007.60096
[4] DOI: 10.1016/0304-4149(94)90113-9 · Zbl 0799.60015
[5] Embrechts P., Modelling Extremal Events for Insurance and Finance (1997) · Zbl 0873.62116
[6] DOI: 10.1016/S0304-4149(99)00030-7 · Zbl 0997.60041
[7] DOI: 10.1016/S0304-4149(00)00083-1 · Zbl 1047.60040
[8] DOI: 10.1080/15326349108807204 · Zbl 0747.60062
[9] Tang Q., Stochastic Process. Appl. 108 pp 299– (2003)
[10] DOI: 10.1023/B:EXTR.0000031178.19509.57 · Zbl 1049.62017
[11] DOI: 10.1239/aap/1103662967 · Zbl 1095.91040
[12] DOI: 10.1360/022004-16
[13] DOI: 10.1080/03461230500361943 · Zbl 1144.91026
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