Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization
Probability
2007-05-23 v3
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].
Cite
@article{arxiv.math/0406392,
title = {Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization},
author = {Rainer Siegmund-Schultze and Heinrich von Weizsaecker},
journal= {arXiv preprint arXiv:math/0406392},
year = {2007}
}
Comments
10 pages, some references added, typos corrected