Local time, upcrossing time and weak cutpoints of a spatially inhomogeneous random walk on the line

TL;DR

This paper analyzes a transient inhomogeneous random walk on the positive line, providing criteria for certain special points related to local time and upcrossings, and establishing their asymptotic distribution as exponential.

Contribution

It introduces new criteria for the finiteness and distribution of special points like weak cutpoints in inhomogeneous random walks, answering three conjectures from prior research.

Findings

Criteria for finiteness of points with equal local/upcrossing times

Order of expected number of such points in [1,n] for certain drifts

Asymptotic distribution of the number of these points converges to exponential

Abstract

In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drifts on the lattice of the positive half line. We give criteria for the finiteness of the number of points having exactly the same local time and/or upcrossing time and weak cutpoints (a point $x$ is called a weak cutpoint if the walk never returns to $x-1$ after its first upcrossing from $x$ to $x+1$). In addition, for the walk with some special local drifts, we also give the order of the expected number of these points in $[1,n].$ Finally, we show that, when properly scaled, the number of these points in $[1,n]$ converges in distribution to a random variable with the standard exponential distribution. Our results answer three conjectures related to the local time, the upcrossing time, and the weak cutpoints proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638].