Randomly weighted sums of subexponential random variables with application to ruin theory.(English)Zbl 1049.62017

Summary: Let $$\{X_k,1\leq k\leq n\}$$ be $$n$$ independent and real-valued random variables with common subexponential distribution function, and let $$\{\theta_k, 1\leq k\leq n\}$$ be other $$n$$ random variables independent of $$\{X_k,1\leq k\leq n\}$$ and satisfying $$a\leq \theta_k \leq b$$ for some $$0<a\leq b<\infty$$ for all $$1\leq k\leq n$$. This paper proves that the asymptotic relations $\mathbb{P}\left(\max_{1\leq m\leq n} \sum^m_{k=1}\theta_k X_k>x\right)\sim\mathbb{P} \left(\sum^n_{k=1} \theta_k X_k >x\right)\sim \sum^p_{k=1}\mathbb{P}(\theta_kX_k>x)$ hold as $$x\to\infty$$. In doing so, no assumption is made on the dependence structure of the sequence $$\{\theta_k,1\leq k\leq n\}$$. An application to ruin theory is proposed.

MSC:

 62E20 Asymptotic distribution theory in statistics 91B30 Risk theory, insurance (MSC2010) 60G50 Sums of independent random variables; random walks
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