Descents following maximal values in samples of geometric random variables

We consider samples of geometric random variables and find the average size of the descent after the first and last maximal values. These are asymptotically but not exactly equal, with the descent after the last maximum being slightly larger than that after the first. Thereafter we calculate the probability that the descent after the last maximum is equal to, greater than, or less than the descent after the first maximum. Finally we compute asymptotic expansions for these probabilities.