On descents after maximal values in samples of discrete random variables

We show that the expected value of the descent after the first maximum in a sample of i.i.d. discrete random variables, as the sample size grows, behaves asymptotically up to vanishing terms as the expectation of the maximal value minus the expectation of a sampled random variable, provided the latter is finite. We also show that the expected value after the last maximum exhibits the same behaviour, although it is in general slightly bigger in mean.