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Tail behavior of negatively associated heavy-tailed sums

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jaap Geluk  and
Kai W. Ng
Jaap Geluk*
Affiliation:
The Petroleum Institute, Abu Dhabi
Kai W. Ng*
Affiliation:
University of Hong Kong
*
∗Postal address: The Petroleum Institute, PO Box 2533, Abu Dhabi, United Arab Emirates. Email address: jgeluk@pi.ac.ae
∗∗Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong. Email address: kaing@hku.hk
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Abstract

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Consider a sequence {Xk, k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S(n) = max0 ≤ k ≤ nSk, with X0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where Xk, k ≥ 1, are not necessarily identically distributed and/or independent.

Keywords

Asymptoticssubexponentialitypartial sumtail probabilitynegative association

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 62E20: Asymptotic distribution theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 587 - 593
Copyright
© Applied Probability Trust 2006 

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