Notes on a certain local time and excursions of simple symmetric random walks

In this article, we deal with the local time of simple symmetric random walks which counts number times that walk goes between zero and one their excursions. First, by depicting typical sample paths time, give a visual proof discrete analogue Lévy’s theorem states relationship maximum walk. Second, excursions in case are introduced an example calculation some probability generating function using marked is demonstrated. Finally, provide arc-sine law through excursion technique for provides distribution total spends on positive side.