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

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 $P(B(T) > x)$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality $P(B(T) > bx) \sim P(M(T) > bx) \sim P(T>x)$ 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.