The geometric convergence rate of a Lindley random walk

Let {X-n} be the Lindley random walk on [0,infinity) defined by X-n=max[Xn-1+A(n),0] for n greater than or equal to 1 with X-0=x greater than or equal to 0. Here, {A(n)} is a sequence of independent and identically distributed random variables. When E[A(1)]<0 and E[r(A1)]<infinity for some r>1, {X-n} converges at a geometric rate in total variation to an invariant distribution pi; i.e. there exists s > 1 such that (n-->infinity)lim s(n) (B)sup \P-x[X-n is an element of B]-pi(B)\=0 for every initial state x greater than or equal to 0. In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is s=phi(r(0))(-1) where phi(r)=E[r(A1)] and r(0)>1 satisfies phi'(r(0))=0.