On the Height of One-Dimensional Random Walk

Consider the one-dimensional random walk Xn: as it evolves (at each unit of time), it either increases by one with probability p or resets to 0 with probability 1-p. In the present paper, we analyze the law of the height statistics Hn, corresponding to our model Xn. Also, we prove that the limiting distribution of the walk Xn is a shifted geometric distribution with parameter 1-p and find the closed forms of the mean and the variance of Xn using the probability-generating function.