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

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].
Comment: In this version, we answer two more problems proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638]. The first version considered only the number of points with exactly the same local time