SPITZERS CONDITION AND LADDER VARIABLES IN RANDOM-WALKS

Spitzer's condition holds for a random walk if the probabilities rho(n) = P{S-n > 0} converge in Cesaro mean to rho, where 0 < rho < 1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that rho(n) converges to rho. This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.