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On Sums of Products of Bernoulli Variables and Random Permutations

Author

Listed:
  • Anatole Joffe

    (Département de Mathématiques et de Statistique)

  • Éric Marchand

    (Department of Mathematics and Statistics)

  • François Perron

    (Département de Mathématiques et de Statistique)

  • Paul Popadiuk

    (Department of Mathematics and Statistics)

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 $$p = \sqrt {\lambda h} + o(1/\sqrt n )$$ .

Suggested Citation

  • Anatole Joffe & Éric Marchand & François Perron & Paul Popadiuk, 2004. "On Sums of Products of Bernoulli Variables and Random Permutations," Journal of Theoretical Probability, Springer, vol. 17(1), pages 285-292, January.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:1:d:10.1023_b:jotp.0000020485.34082.8c
    DOI: 10.1023/B:JOTP.0000020485.34082.8c
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    References listed on IDEAS

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    1. Ling, K. D., 1988. "On binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 6(4), pages 247-250, March.
    2. Demetrios Antzoulakos & Stathis Chadjiconstantinidis, 2001. "Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 599-619, September.
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