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: $Y(t) = X(t) - \inf_{0\leq s\leq t}X(s)$, $M(t,x)=\sup_{0\leq s\leq t}Y(s)-x$ 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 $t\te{-\gamma x}$ has a positive limit ($\gamma$ denotes the Cram\'er coefficient), our results together with the theorem of Doney & Maller (2005) imply the existence and the explicit form of the joint weak limit $(Y(\infty),M(\infty),Z(\infty))$.