On the number of crossings of a strip by sample paths of a random walk

Exact expressions are obtained for the distribution of the total number of crossings of a strip by sample paths of a random walk whose jumps have a two-sided geometric distribution. The distribution of the number of crossings during a finite time interval is found in explicit form for walks with jumps taking the values $\pm1$. A limit theorem is proved for the joint distribution of the number of crossings of an expanding strip on a finite (increasing) time interval and the position of the walk at the end of this interval, and the corresponding limit distribution is found.