Let {Xn} be the Lindley random walk on [0,∞) defined by Xn=max[Xn-1+An, 0] for n ≧ 1 with X0 = x ≧ 0. Here, {An} is a sequence of independent and identically distributed random variables. When E[A1] < 0 and E[rA1] < ∞ for some r > 1, {Xn} converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s > 1 such that limn→x Sn supB | Px[Xn ∈ B] – π(B) | = 0 for every initial state x ≧ 0. In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels: the largest s satisfying the above relationship is s = φ(r0)-1 where φ(r) = E[rA1] and r0 > 1 satisfies φ′(r0) = 0.
