Abstract
Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z = XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.
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Research supported by National Natural Science Foundation of China (No. 11071182, 11001052), China Postdoctoral Science Foundation (No. 20100471365), Natural Science Foundation of Jiangsu Province of China (No. BK2010480), Postdoctoral Research Program of Jiangsu Province of China (No. 0901029C), Jiangsu Government Scholarship for Overseas Studies, Qing Lan Project.
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Yang, Y., Wang, Y. Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16, 55–74 (2013). https://doi.org/10.1007/s10687-012-0153-2
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DOI: https://doi.org/10.1007/s10687-012-0153-2
Keywords
- Bivariate Sarmanov distribution
- Product
- Regular variation
- Finite-time and infinite-time ruin probabilities