Joint asymptotic distribution of certain path functionals of the reflected process

Abstract

Let $\tau (x)$ be the first time that the reflected process $Y$ of a Lévy process $X$ crosses $x>0$. The main aim of this paper is to investigate the joint asymptotic distribution of $Y(t)=X(t)-\underset{0\le s\le t}{inf}X(s)$ and the path functionals $Z(x)=Y(\tau (x))-x$ and $m(t)=\underset{0\le s\le t}{sup}Y(s)-y^{\ast }(t)$, for a certain non-linear curve $y^{\ast }(t)$. We restrict to Lévy processes $X$ satisfying Cramér’s condition, a non-lattice condition and the moment conditions that $E[|X(1)|]$ and $E[\exp (\gamma X(1))|X(1)|]$ are finite (where $\gamma$ denotes the Cramér coefficient). We prove that $Y(t)$ and $Z(x)$ are asymptotically independent as $min\{t,x\}\to \mathrm{\infty }$ and characterise the law of the limit $(Y_{\mathrm{\infty }},Z_{\mathrm{\infty }})$. Moreover, if $y^{\ast }(t)={\gamma }^{-1}\log (t)$ and $min\{t,x\}\to \mathrm{\infty }$ in such a way that $t\exp \{-\gamma x\}\to 0$, then we show that $Y(t)$, $Z(x)$ and $m(t)$ are asymptotically independent and derive the explicit form of the joint weak limit $(Y_{\mathrm{\infty }},Z_{\mathrm{\infty }},m_{\mathrm{\infty }})$. The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law $(Y_{\mathrm{\infty }},Z_{\mathrm{\infty }})$.