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Probability set functions

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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.

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Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Mail Stop K710, 87545, Los Alamos, New Mexico, USA

    William J. Bruno, Gian-Carlo Rota & David C. Torney

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  1. William J. Bruno
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  2. Gian-Carlo Rota
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  3. David C. Torney
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Additional information

Dedicated to our friend, W.A. Beyer. Financial support for this work was derived from the U.S.D.O.E. Human Genome Project, through the Center for Human Genome Studies at Los Alamos National Laboratory, and also through the Center for Nonlinear Studies, Los Alamos National Laboratory, LANL report LAUR-97-323.

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Bruno, W.J., Rota, GC. & Torney, D.C. Probability set functions. Annals of Combinatorics 3, 13–25 (1999). https://doi.org/10.1007/BF01609871

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  • DOI: https://doi.org/10.1007/BF01609871

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