Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks
Abstract
In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the sense of Mauldin, Monticino and v.Weizsaecker [1] is transient. On the other hand we construct an example of a transient but polygonally recurrent random walk with independent components on Z^2.
Cite
@article{arxiv.math/0406423,
title = {Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks},
author = {Rainer Siegmund-Schultze and Heinrich von Weizsaecker},
journal= {arXiv preprint arXiv:math/0406423},
year = {2007}
}
Comments
23 pages, errors and typos corrected, references added