## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 43, 18 pp.

### Joint asymptotic distribution of certain path functionals of the reflected process

Aleksandar Mijatović and Martijn Pistorius

#### 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 $Y(t)=X(t) - \inf _{0\leq s\leq t}X(s)$ 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 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 [7] and our characterisation of the law $(Y_\infty , Z_\infty )$.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 43, 18 pp.

**Dates**

Received: 12 June 2015

Accepted: 9 May 2016

First available in Project Euclid: 23 May 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1464033418

**Digital Object Identifier**

doi:10.1214/16-ECP4359

**Mathematical Reviews number (MathSciNet)**

MR3510251

**Zentralblatt MATH identifier**

1346.60062

**Subjects**

Primary: 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

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Mijatović, Aleksandar; Pistorius, Martijn. Joint asymptotic distribution of certain path functionals of the reflected process. Electron. Commun. Probab. 21 (2016), paper no. 43, 18 pp. doi:10.1214/16-ECP4359. https://projecteuclid.org/euclid.ecp/1464033418