Local time of transient random walk on the lattice of positive half line

In this paper, we study spatially inhomogeneous random walks on the lattice of positive half line. Fix a positive integer $a.$ Let $C$ be the collection of points at which the local times of the random walk are exactly $a.$ We give a criterion to tell whether $C$ is a finite set or not. Our result answer an open problem proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638.]. When $a=1,$ $C$ is just the set of strong cutpoints studied in the above mentioned paper. By a moment method, when the local drift of the walk at $i\ge 1$ is $1/i,$ we show that $2|C\cap [0,n]|/\log n\overset{\mathcal D}{\rightarrow} S$ as $n\rightarrow\infty,$ where $S$ is an exponentially distributed random variable with $P(S>t)=e^{-t},\ t>0.$
Comment: 25 pages