Factorial moments and enumerating distributions

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Abstract
Summary. It is shown that, if a linear stochastic variable attains, with probability 1, non-negative values only, and if its distribution function (which need not be an enumerating distribution function) has moments of arbitrarily high order, then the sequence of these moments can be represented as the sequence of the factorial moments of an enumerating distribution. The latter is obtained as a stratification of Poissonian distributions, namely, by a Stieltjes integration of the (variable) standard deviation of Poisson's enumerating distribution function.

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