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Pólya sequences, binomial convolution and the union of random sets

Published online by Cambridge University Press:  14 July 2016

David W. Walkup
David W. Walkup*
Affiliation:
Washington University, St. Louis
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Abstract

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.

Keywords

PÓLYA FREQUENCY SEQUENCETOTAL POSITIVITYCONVOLUTIONBINOMIAL COEFFICIENTSEXPONENTIAL GENERATING FUNCTIONSRELIABILITYINCREASING FAILURE RATE
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 76 - 85
Copyright
Copyright © Applied Probability Trust 1976 

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References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford.Google Scholar
Walkup, D. W. (1974) Matchings in random regular bipartite digraphs. Department of Computer Science, Washington University, St. Louis, Report CS74–10.Google Scholar