Joint asymptotic distribution of certain path functionals of the reflected process. (English) Zbl 1346.60062

Mijatović, Aleksandar; Pistorius, Martijn

Electron. Commun. Probab. 21, Paper No. 43, 18 p. (2016).

Summary: 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 \(Y(t)=X(t) - \inf _{0\leq s\leq t}X(s)\) and the path functionals \(Z(x)=Y(\tau (x))-x\) and \(m(t)=\sup _{0\leq s\leq t}Y(s) - y^*(t)\), for a certain non-linear curve \(y^*(t)\). We restrict ourselves 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 \(\min \{t,x\}\to \infty \) 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 \(t\exp \{-\gamma x\}\to 0\), 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 excursion theory, Theorem 1 in [R. A. Doney and R. A. Maller, Ann. Appl. Probab. 15, No. 2, 1445–1450 (2005; Zbl 1069.60045)] and our characterisation of the law \((Y_\infty , Z_\infty )\).

MSC:

60G51	Processes with independent increments; Lévy processes
60F05	Central limit and other weak theorems
60G17	Sample path properties

Keywords:

reflected Lévy process; asymptotic independence; limiting overshoot; Cramér condition

Citations:

Zbl 1069.60045