### On runs, bivariate Poisson mixtures and distributions that arise in Bernoulli arrays

**Éric Marchand**

*(Université de Sherbrooke)*

**Djilali Ait Aoudia**

*(Université du Québec à Montréal)*

**François Perron**

*(Université de Montréal)*

**Latifa Ben Hadj Slimene**

*(Université de Sherbrooke)*

#### Abstract

Distributional findings are obtained relative to various quantities arising in Bernoulli arrays $\{ X_{k,j}, k \geq 1, j =1, \ldots, r+1\}$, where the rows $(X_{k,1}, \ldots, X_{k,r+1})$ are independently distributed as $\hbox{Multinomial}(1,p_{k,1}, \ldots,p_{k,r+1})$ for $k \geq 1$ with the homogeneity across the first $r$ columns assumption $p_{k,1}= \cdots = p_{k,r}$. The quantities of interest relate to the measure of the number of runs of length $2$ and are $\underline{S}_n (S_{n,1}, \ldots, S_{n,r})$, $\underline{S}=\lim_{n \to \infty} \underline{S}_n$, $T_n=\sum_{j=1}^r S_{n,j}$, and $T=\lim_{n \to \infty} T_n$, where $S_{n,j}= \sum_{k=1}^n X_{k,j} X_{k+1,j}$. With various known results applicable to the marginal distributions of the $S_{n,j}$'s and to their limiting quantities $S_j=\lim_{n \to \infty} S_{n,j}\,$, we investigate joint distributions in the bivariate ($r=2$) case and the distributions of their totals $T_n$ and $T$ for $r \geq 2$. In the latter case, we derive a key relationship between multivariate problems and univariate ($r=1$) problems opening up the path for several derivations and representations such as Poisson mixtures. In the former case, we obtain general expressions for the probability generating functions, the binomial moments and the probability mass functions through conditioning, an analysis of a resulting recursive system of equations, and again by exploiting connections with the univariate problem. More precisely, for cases where $p_{k,j}= \frac{1}{b+k}$ for $j=1,2$ with $b \geq 1$, we obtain explicit expressions for the probability generating function of $\underline{S}_n$, $n \geq 1$, and $\underline{S}$, as well as a Poisson mixture representation : $\underline{S}|(V_1=v_1, V_2=v_2) \sim^{ind.} \mbox{Poisson}(v_i)$ with $(V_1,V_2) \sim \mbox{Dirichlet}(1,1,b-1)$ which nicely captures both the marginal distributions and the dependence structure. From this, we derive the fact that $S_1|S_1+S_2=t$ is uniformly distributed on $\{0,1,\ldots,t\}$ whenever $b=1$. We conclude with yet another mixture representation for $p_{k,j}= \frac{1} {b+k}$ for $j=1,2$ with $b \geq 1$, where we show that $\underline{S}|\alpha \sim p_{\alpha}$, $\alpha \sim \hbox{Beta}(1,b)$ with $p_{\alpha}$ a bivariate mass function with Poisson$(\alpha)$ marginals given by $p_{\alpha} (s_1,s_2)= \frac{e^{-\alpha} {\alpha}^{s_1+s_2}} {(s_1+s_2+1)!} \, (s_1+s_2+1-\alpha)\,.$

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Pages: 1-12

Publication Date: February 15, 2014

DOI: 10.1214/ECP.v19-3152

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