The maximal height of simple random walks revisited

Erosion by a one-dimensional random walk

“…and corresponding results of Katzenbeisser and Panny [20] for the rightmost site visited in the ordinary Pólya walk, yielding for one-sided erosion…”

On the Height of One-Dimensional Random Walk

“…Furthermore, we studied the statistical properties of the random walk X n like the limit distribution, the mean, and the variance. Firstly, we found the closed form of the probability-generating function of the random walk X n from the recursive equation defined in Equation (10). Next, we proved that the limiting distribution of X n is a shifted geometric distribution with parameter 1 − p. Finally, we derived the probability-generating function of X n to obtain the closed forms of the mean and the variance of X n .…”

“…Now, we use Equation (10) to show that the random walk X n converges to a shifted geometric distribution with parameter 1 − p asymptotically. It is introduced in the next theorem.…”

“…Also, Csaki and Hu analyzed the lengths and heights of random walk excursions in [9]. Furthermore, Katzenbeisser and Panny studied the maximal height of simple random walks, which were revisited in [10]. In addition, Banderier and Nicodème [11] studied the height of discrete bridges/meanders/excursions for bounded discrete walks.…”