Tail Asymptotics for Cumulative Processes Sampled at Heavy-Tailed Random Times with Applications to Collective Risk and Queueing Models in Markovian Environment

Details Tail Asymptotics for Cumulative Processes Sampled at Heavy-Tailed Random Times with Applications to Collective Risk and Queueing Models in Markovian Environment Tail Asymptotics for Cumulative Processes Sampled at Heavy-Tailed Random Times with Applications to Collective Risk and Queueing Models in Markovian Environment

2012-03-29

arxiv.org/abs/1203.6574v1

Mathematics

Probability

This is a preprint of a paper submitted for publication in "Stochastic Models". This preprint includes the proofs of the lemma

Scientific paper

This paper studies the tail asymptotics for a cumulative process $\{B(t); t \ge 0\}$ sampled at heavy-tailed random times $T$, where $T$ has a dominant impact on the asymptotic behavior of $\PP(B(T) > x)$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality $\PP(B(T) > bx) \sim \PP(M(T) > bx) \sim \PP(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. Their typical applications are collective risk models with Markov-correlated claim sizes, and queueing models with batch Markovian arrival process.

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