Long Excursions of a Random Walk

In Csaki et al. (1) and Revesz and Willekens(9) it was proved that the length of the longest excursion among the first n excursions of a plane random walk is nearly equal to the total sum of the lenghts of these excursions. In this paper several results are proved in the same spirit, for plane random walks and for random walks in higher dimensions.