On a class of reflected AR(1) processes

In this paper we study a reflectedAR(1) process, i.e. a process (Zn)n obeying the recursion Z_n+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 Yn - Bn, 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.