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

Abstract

In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drift 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 drift, we also give the order of the expected number of these points in $[1,n].$ Finally, if the local drift at $n$ is $\frac{{\rm Y}}{2n}$ with ${\rm Y}>1$ for $n$ large enough, we show that, when properly scaled the number of these points in $[1,n]$ converges in distribution to a random variable with Gamma$({\rm Y}-1,1)$ distribution. Our results answer three conjectures related to the local time, the upcrossing time, and the weak cutpoints posed by E. Csáki, A. Földes, P. Révész [J. Theoret. Probab. 23 (2) (2010) 624-638].