Conditions for recurrence and transience for one family of random walks

TL;DR

This paper investigates the recurrence and transience conditions of a specific parametric family of correlated two-dimensional random walks confined to the quarter plane, providing precise criteria based on the parameter a.

Contribution

The paper derives explicit conditions on the parameter a that determine when the random walk is recurrent, advancing understanding of correlated walks in constrained domains.

Findings

Derived recurrence conditions for the family of random walks.

Identified parameter ranges leading to transience or recurrence.

Provided a framework for analyzing correlated two-dimensional walks.

Abstract

A parametric family of two-dimensional random walks $\mathbf{S}_t(a)$ $=\big(S_t^{(1)}(a),$ $S_t^{(2)}(a)\big)$ in the main quarter plane is studied. The components $S_t^{(1)}(a)$ and $S_t^{(2)}(a)$ are assumed to be correlated in the way that is defined exactly in the paper. We derive the conditions on $a$, under which a random walk $\mathbf{S}_t(a)$ is recurrent.