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