Spitzer’s condition and ladder variables in random walks. (English) Zbl 0818.60060

Doney, R. A.

Probab. Theory Relat. Fields 101, No. 4, 577-580 (1995).

Summary: Spitzer’s condition holds for a random walk if the probabilities \(\rho_ n = P\{S_ n > 0\}\) converge in Cèsaro mean to \(\rho\), where \(0 < \rho < 1\). We answer a question which was posed both by F. Spitzer [Trans. Am. Math. Soc. 82, 323-339 (1956; Zbl 0071.130)] and by D. J. Emery [Z. Wahrscheinlichkeitstheorie Verw. Geb. 31, 125-139 (1975; Zbl 0283.60070)] 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 of the author and P. E. Greenwood [Probab. Theory Relat. Fields 94, No. 4, 457-472 (1993; Zbl 0791.60058)] 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.

MSC:

60G50

Sums of independent random variables; random walks

Keywords:

fluctuation theory; Spitzer’s condition; random walk; domain of attraction

Citations:

Zbl 0283.60070; Zbl 0071.130; Zbl 0295.60048; Zbl 0791.60058

References:

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