On a class of reflected AR(1) processes

Abstract

In this paper we study a reflected AR(1) process, i.e. a process (Z_n)_n obeying the recursion Z_n+1= max{aZ_n+X_n,0}, with (X_n)_n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z_n (in terms of transforms) in case X_n can be written as Y_n−B_n, with (B_n)_n being a sequence of independent random variables which are all Exp(λ) distributed, and (Y_n)_n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B_n)_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.