Crossings states and sets of states in Pólya random walks
Abstract
We consider the Pólya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up and downdirected statecrossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected statecrossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$dimensional random walks, $d\geq2$, in bounded areas.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.04218
 Bibcode:
 2015arXiv151204218A
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 60G50;
 60J80;
 60C05;
 60K25
 EPrint:
 Dear readers. I made a tremendous work to revise this paper after referee report. There are 30 pages of 11pt format, 4 figures and 1 table