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 down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in {\mathbb{Z}}^2\setminus \{0\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\ge 2$, in bounded areas.

This paper is accepted for publication in "Methodology and Computing in Applied Probability"

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