Descents following maximal values in samples of geometric random variables

On descents after maximal values in samples of discrete random variables

“…2 Subtracting (11) from (12) gives s m (1 − r n−1 m ). We claim that for all n ≥ 1 this remainder converges to 0 as m → ∞.…”

“…Comparison of the mean descents after the first and the last maxima8 InArchibald et al (2015) it is mentioned that the mean of the difference between descents after the first and the last 9 maxima can be found but the explicit expression is lengthy. In a related work(Blecher et al, 2014) concerning random com-10 positions it is shown by a direct bijection that the mean descent after the last maximum is greater than the mean descent 11 after the first maximum in a random composition.…”

“…We show that for any sample of i.i.d. N-valued random variables this is12 always the case, unless one defines Z h to explicitly depend on whether the maximum at the last position in the sample is13 the first maximum or not. To exclude the latter possibility suppose that14 …”