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Weighted sums of subexponential random variables and their maxima

Part of:Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Yiqing Chen,
Kai W. Ng and
Qihe Tang
Yiqing Chen*
Affiliation:
Guangdong University of Technology
Kai W. Ng*
Affiliation:
The University of Hong Kong
Qihe Tang*
Affiliation:
Concordia University
*
∗ Postal address: School of Economics and Management, Guangdong University of Technology, East Dongfeng Road 729, Guangzhou 510090, Guangdong, P. R. China. Email address: yiqch@263.net
∗∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: kaing@hku.hk
∗∗∗ Postal address: Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, Canada. Email address: qtang@mathstat.concordia.ca
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Abstract

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Let {X k , k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {w k , k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1 n w k X k and the maximum of weighted sums max1≤m≤n ∑k=1 m w k X k , subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss X k during the kth year, with a discount or inflation factor w k .

Keywords

AsymptoticsMatuszewska indexweighted summaximumsubexponentialitytail probabilityuniformityruin probabilitydiscounted loss

MSC classification

Secondary:60G50: Sums of independent random variables; random walks60G70: Extreme value theory; extremal processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005, pp. 510 - 522
Copyright
Copyright © Applied Probability Trust 2005 

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