Summary
Spitzer's condition holds for a random walk if the probabilities ρ n =P{ n > 0} converge in Cèsaro mean to ϱ, where 0<ϱ<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 ρn converges to ϱ. 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.
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Doney, R.A. Spitzer's condition and ladder variables in random walks. Probab. Th. Rel. Fields 101, 577–580 (1995). https://doi.org/10.1007/BF01202785
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DOI: https://doi.org/10.1007/BF01202785