Spitzer's condition and ladder variables in random walks

Summary Spitzer's condition holds for a random walk if the probabilities _n =P{ _n > 0} converge in Cèsaro mean to , where 0<<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 n converges to . 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.