It is not hard to see that, for r > 2, the r-th factorial moment of Xn is equal to the expected number of ordered r-tuples (G1,.
For every integer r > 2, the r-th factorial moment of [T.
Also, the nth factorial moment of X with mean [lambda], denoted by [E.
We note that identities (42), (43), and (44) can be shown to be particular cases of the generalized factorial moments by Mangontarum and Corcino  given by
When the moments of the mixing variable exist, a necessary and sufficient condition that a distribution be a mixed Poisson is that its factorial moment (3) generating function be equal to the moment-generating function of the mixing distribution (Haight, 1967), namely
k] denote respectively the kth factorial moment of the mixed Poisson variable and the kth moment about zero of the mixing variable.
Theorem 3 The expectation and the second factorial moment of the number of superior elements that we reject when we consider the consecutive maxima instead of the true maxima, assuming random words of length n, are given by
For the second factorial moment, things are more involved, and we must find the continuation of
The cancellations leave us with a second factorial moment of
To do this we need to look at two possible sources, namely fluctuations occurring in the second factorial moment calculations and those fluctuations from the expected value which, when squared, will involve an [n.
X is said to be smaller than Y in factorial moments
ordering, denoted by X [[less than or equal to].
This requires of course a very detailed description of the asymptotic behaviour of the ordinary or factorial moments