Abstract. Let $\tau(x)$ be the first time that the reflected process $Y$ of a L\'evy process $X$ crosses $x>0$. The main aim of this paper is to investigate the joint asymptotic distribution of the path functionals $Y(t) = X(t) - \inf_{0\leq s\leq t}X(s)$, $Z(x)=Y(\tau(x))-x$, and $m(t)=\sup_{0\leq s\leq t}Y(s) - y^*(t)$ for a certain non-linear curve $y^*(t)$. We prove that under Cram\'{e}r's condition on $X(1)$ the functionals $Y(t)$ and $Z(x)$ are asymptotically independent as $\min\{t,x\}\to\infty$ and characterise the law of the limit $(Y(\infty),Z(\infty))$. Moreover, if $y^*(t) = \gamma^{-1}\log(t)$ and $\min\{t,x\}\to\infty$ in such a way that $t\exp\{-\gamma x\}\to 0$ ($\gamma$ denotes the Cram\'er coefficient), then we show that $Y(t)$, $Z(x)$ and $m(t)$ are asymptotically independent and derive the explicit form of the joint weak limit $(Y(\infty), Z(\infty), m(\infty))$. The proof is based on the theorem of Doney \& Maller~\cite{DoneyMaller} together with our characterisation of the law $(Y(\infty), Z(\infty))$.

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