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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相依随机变量乘积布雷曼定理的扩展及其在破产理论中的应用

我们考虑两个因变量X和\（\ Theta \）乘积的尾部行为。受Denisov和Zwart（J Appl Probab 44：1031-1046，2007）的激励，我们放松了Breiman定理中现有\（\ alpha \，+ \，\ epsilon \）时刻\（\ Theta \）的条件。到现有的\（\ alpha \）时刻，并获得与从属产品\（X \ Theta \）的Breiman定理相似的结果，而X和\（\ Theta \）遵循copula函数。作为应用程序，我们考虑具有相关保险和金融风险的离散时间保险风险模型，并得出（无限）有限破产概率的渐近尾部行为。