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Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory

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Abstract

We consider the tail behavior of the product of two dependent random variables X and \(\Theta \). Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing \(\alpha \,+\,\epsilon \) th moment of \(\Theta \) in Breiman’s theorem to the existing \(\alpha \)th moment and obtain the similar result as Breiman’s theorem of the dependent product \(X \Theta \), while X and \(\Theta \) follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

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Acknowledgements

The authors are grateful to the editor and the anonymous referees for their helpful comments and suggestions, which have helped us produce a substantially improved version. The work is supported by the National Key Research and Development Plan (No. 2016YFC0800104) and NSFC (No. 71771203).

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Authors and Affiliations

  1. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yu Chen, Dan Chen & Wenxue Gao

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  1. Yu Chen
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  2. Dan Chen
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  3. Wenxue Gao
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Correspondence to Yu Chen.

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Chen, Y., Chen, D. & Gao, W. Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory. Commun. Math. Stat. 7, 1–23 (2019). https://doi.org/10.1007/s40304-018-0132-2

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  • DOI: https://doi.org/10.1007/s40304-018-0132-2

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