Geodesic
On the number of crossings of a~strip by sample paths of a~random walk
Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 927-939

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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. 
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     author = {V. I. Lotov and N. G. Orlova},
     title = {On the number of crossings of a~strip by sample paths of a~random walk},
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V. I. Lotov; N. G. Orlova. On the number of crossings of a~strip by sample paths of a~random walk. Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 927-939. http://geodesic.mathdoc.fr/item/SM_2003_194_6_a7/