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On Sums of Products of Bernoulli Variables and Random Permutations

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Abstract

Let {X k } k≥1 be independent Bernoulli random variables with parameters p k . We study the distribution of the number or runs of length 2: that is \(S_n = \sum {_{k = 1}^n {\text{ }}X_k X_{k + 1}}\). Let S=lim n→∞ S n . For the particular case p k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p k =p for all k we obtain the generating function of S n and the limiting distribution of S n for \(p = \sqrt {\lambda h} + o(1/\sqrt n )\).

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Authors and Affiliations

  1. Département de Mathématiques et de Statistique, C.P. 6128, Succursale Centre-Ville, Montréal, Québec, Canada, H3C 3J7

    Anatole Joffe & François Perron

  2. Department of Mathematics and Statistics, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3

    Éric Marchand

  3. Department of Mathematics and Statistics, 7141 Sherbrooke St. West, Montréal, Québec, Canada, H4B 1R6

    Paul Popadiuk

Authors
  1. Anatole Joffe
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  2. Éric Marchand
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  3. François Perron
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  4. Paul Popadiuk
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Correspondence to François Perron.

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Joffe, A., Marchand, É., Perron, F. et al. On Sums of Products of Bernoulli Variables and Random Permutations. Journal of Theoretical Probability 17, 285–292 (2004). https://doi.org/10.1023/B:JOTP.0000020485.34082.8c

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  • DOI: https://doi.org/10.1023/B:JOTP.0000020485.34082.8c

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