## Journal of Applied Probability

### On a class of reflected AR(1) processes

#### Abstract

In this paper we study a reflected AR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1= max{aZn+Xn,0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as YnBn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

#### Article information

Source
J. Appl. Probab., Volume 53, Number 3 (2016), 818-832.

Dates
First available in Project Euclid: 13 October 2016

https://projecteuclid.org/euclid.jap/1476370778

Mathematical Reviews number (MathSciNet)
MR3570096

Zentralblatt MATH identifier
1351.60121

#### Citation

Boxma, Onno; Mandjes, Michel; Reed, Josh. On a class of reflected AR(1) processes. J. Appl. Probab. 53 (2016), no. 3, 818--832. https://projecteuclid.org/euclid.jap/1476370778

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