On Sums of Products of Bernoulli Variables and Random Permutations

Let {Xk}k≥1 be independent Bernoulli random variables with parameters pk. We study the distribution of the number or runs of length 2: that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$S_n = \sum {_{k = 1}^n {\text{ }}X_k X_{k + 1}}$$ \end{document}. Let S=limn→∞Sn. For the particular case pk=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 pk=p for all k we obtain the generating function of Sn and the limiting distribution of Sn for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$p = \sqrt {\lambda h} + o(1/\sqrt n )$$ \end{document}.