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

Tang, Qihe; Tsitsiashvili, Gurami

Extremes 6, No. 3, 171-188 (2003).

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

Keywords:

asymptotics; dominated variation; ruin probability; subexponentiality; uniformity; heavy tailed distributions