The Maximal Height of Simple Random Walks Revisited

In a recent paper Katzenbeisser and Panny (1996) derived distributional results for a number of so called simple random walk statistics defined on a simple random walk in the sense of Cox and Miller (1968) starting at zero and leading to state 1 after n steps, where 1 is arbitrary, but fix. In the present paper the random walk statistics Dn = the one-sided maximum deviation and Qn = the number of times where the maximum is achieved, are considered and distributional results are presented, when it is irrespective, where the random walk terminates after n steps. Thus, the results can be seen as generalizations of some well known results about (purely) binomial random walk, given e.g. in Revesz (1990).