On the number of maxima in a discrete sample

Let Mn = max(N1,..., Nn), where N1, N2, ... are i.i.d., positive, integer-valued r.v.'s. We are interested in Kn, the number of values of j [epsilon] {1, 2, ..., n} for which Nj = Mn, especially for large values of n. There is strong evidence that Kn either tends to one or to infinity, or diverges in distribution as n tends to infinity. An interesting example of the latter type occurs when N1 has a geometric distribution. There is an application of results on Kn to the behaviour of the fractional parts of sample maxima from non-integer populations.