On a class of reflected AR(1) processes

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(lambda) distributed, and (Y-n)(n) i.i.d.; when vertical bar a vertical bar < 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.