Joint asymptotic distribution of certain path functionals of the reflected process

Abstract

Let $\tau (x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t)=X(t)-{\inf }_{0\le s\le t}X(s)$, $M(t,x)={\sup }_{0\le s\le t}Y(s)-x$ and $Z(x)=Y(\tau (x))-x$. We prove that under Cramer's condition on X(1), the functionals $Y(t)$, $M(t,y)$ and $Z(x+y)$ are asymptotically independent as $\min \{t,y,x\}\to \infty$. We also characterise the law of the limiting overshoot $Z(\infty )$ of the reflected process. If, as $\min \{t,x\}\to \infty$, the quantity $t\te{-γx}$ has a positive limit ($\gamma$ denotes the Cramér coefficient), our results together with the theorem of Doney & Maller (2005) imply the existence and the explicit form of the joint weak limit $(Y(\infty ),M(\infty ),Z(\infty ))$.

21 pages, 1 figure

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