State-crossings in multidimensional random walks

Vyacheslav M. Abramov

Published 2015-12-14Version 1

We consider the classic $d$-dimensional symmetric random walk in $\mathbb{Z}^d$ assuming that it starts from the origin and after some excursion returns to the original point again. Under this assumption, the paper established inequalities for the conditional probability distributions of the number of usual (undirected) and specifically defined in the paper below- and above-directional state-crossings. This type of problem for random walks has not been previously considered in the literature, and the results obtained in this paper are pioneering.