Annals of Combinatorics 3 (1999) 13-25

Probability Set Functions

William J. Bruno1, Gian-Carlo Rota, and David C. Torney1

1Theoretical Division, Mail Stop K710, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
 {billb, dct}

Received January 6, 1999

AMS Subject Classification: 60C05, 60G99

Abstract. A probability set function is interpretable as a probability distribution on binary sequences of fixed length. Cumulants of probability set functions enjoy particularly simple properties which make them more manageable than cumulants of general random variables. We derive some identities satisfied by cumulants of probability set functions which we believe to be new. Probability set functions may be expanded in terms of their cumulants. We derive an expansion which allows the construction of examples of probability set functions whose cumulants are arbitrary, restricted only by their absolute values. It is known that this phenomenon cannot occur for continuous probability distributions. Some particular examples of probability set functions are considered, and their cumulants are computed, leading to a conjecture on the upper bound of the values of cumulants. Moments of probability set functions determined by arithmetical conditions are computed in a final example.

Keywords: algebraic enumeration, binary sequences, set partitions into even-size blocks, restricted set partitions, finite residue classes modulo three, moment, centered moment, cumulant, bounds, Fourier analysis, Möbius inversion.

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