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A necessary and sufficient condition for the subexponentiality of the product convolution

Part of: Distribution theory Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  20 March 2018

Hui Xu ,
Fengyang Cheng ,
Yuebao Wang  and
Dongya Cheng
Hui Xu*
Affiliation:
Soochow University
Fengyang Cheng*
Affiliation:
Soochow University
Yuebao Wang*
Affiliation:
Soochow University
Dongya Cheng*
Affiliation:
Soochow University
*
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
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Abstract

Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY, called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of the product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F, given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

Keywords

Necessary and sufficient conditionproduct convolutionsubexponential distributiondiscrete-time insurance risk model

MSC classification

Primary: 60E05: Distributions
Secondary: 62E20: Asymptotic distribution theory 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 57 - 73
Copyright
Copyright © Applied Probability Trust 2018 

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