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.

In particular, the jth order of the factorial moment of the negative binomial NB(k,p), [[mu].

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

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.

The cancellations leave us with a second factorial moment of

X is said to be smaller than Y in

factorial moments ordering, denoted by X [[less than or equal to].