Spitzer's condition for random walks and Levy processes

Spitzer's condition holds for a random walk S if the probabilities rho(n) = P{S-n > 0} converge in Cesaro mean to rho, and for a Levy process X at infinity (at 0, respectively) if t(-1) integral(0)(t) rho(s)ds --> rho as t --> infinity(0), where rho(s) = P{X(s) > 0}. It has been shown in Doney [4] that if 0 < rho < 1 then this happens for a random walk if and only if rho(n) converges to rho. We show here that this result extends to the cases rho = 0 and rho = 1, and also that Spitzer's condition holds for a Levy process at infinity(0) if and only if rho(t) --> rho as t --> infinity(0).