Joint asymptotic distribution of certain path functionals of the reflected process

Abstract

Let τ(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)−inf0≤s≤tX(s) and the path functionals Z(x)=Y(τ(x))−x and m(t)=sup0≤s≤tY(s)−y*(t) for a certain non-linear curve y*(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(γX(1))|X(1)|] are finite (where γ denotes the Cramér coefficient). We prove that Y(t) and Z(x) are asymptotically independent as min{t,x}→∞ and characterise the law of the limit (Y∞,Z∞). Moreover, if y*(t)=γ−1log(t) and min{t,x}→∞ in such a way that texp{−γx}→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∞,Z∞,m∞). The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law (Y∞,Z∞).