Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization

2004/06/20 by Siegmund-Schultze, Rainer, von Weizsaecker, Heinrich
#60G51 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0406392

Abstract

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for independent identically distributed X,Y is used. In part II we shall discuss the connection of this result to 'polygonal recurrence' of higher-dimensional walks and some conjectures on directionally random walks in the sense of Mauldin, Monticino and v.Weizsaecker [5].

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