Hitting time of a half-line by a two-dimensional nonsymmetric random walk

TL;DR

This paper analyzes the asymptotic probability that a two-dimensional nonsymmetric random walk starting at the origin never hits a specified half-line before time n, under certain moment and aperiodicity conditions.

Contribution

It provides an asymptotic estimate for the hitting probability of a half-line by a 2D nonsymmetric random walk, based on the characteristic function near zero.

Findings

Derived asymptotic estimates for large n

Established conditions on moments and aperiodicity

Linked behavior of characteristic function to hitting probabilities

Abstract

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line $ (- \infty,0] \times {0}$ before time $n$. Let $X^{(1)}=(X_{1},X_{2})$ be the increment of the two-dimensional random walk. For an aperiodic random walk with moment conditions ($E[X_{2}]=0 $ and $ E[|X_{1}|^{\delta}]<\infty, E[|X_{2}|^{2+ \delta}]< \infty $ for some $ \delta \in (0,1)$), we obtain an asymptotic estimate (as $n \rightarrow \infty $) of this probability by assuming the behavior of the characteristic function of $X_{1}$ near zero.