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Precise large deviations for sums of random variables with consistently varying tails

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Kai W. Ng ,
Qihe Tang ,
Jia-An Yan  and
Hailiang Yang
Kai W. Ng*
Affiliation:
University of Hong Kong
Qihe Tang*
Affiliation:
University of Amsterdam
Jia-An Yan*
Affiliation:
Chinese Academy of Sciences, Beijing
Hailiang Yang*
Affiliation:
University of Hong Kong
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong.
∗∗∗ Postal address: Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands.
∗∗∗∗ Postal address: Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China.
∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong.
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Abstract

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Keywords

Consistently varying taildoubly stochastic processheavy tailMatuszewska indexnegative associationprecise large deviationsrandom sums

MSC classification

Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 93 - 107
Copyright
Copyright © Applied Probability Trust 2004 

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