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 $p_{\alpha} (s_1,s_2)= \frac{e^{-\alpha} {\alpha}^{s_1+s_2}} {(s_1+s_2+1)!} \, (s_1+s_2+1-\alpha)\,.$

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-12

Publication Date: February 15, 2014

DOI: 10.1214/ECP.v19-3152

References

  • Ait Aoudia, D. and Marchand É. (2014). On a simple construction of bivariate probability functions with fixed marginals. Technical report 136 URL.
  • Ait Aoudia, Djilali; Marchand, Éric. On the number of runs for Bernoulli arrays. J. Appl. Probab. 47 (2010), no. 2, 367--377. MR2668494
  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 (1992), no. 3, 519--535. MR1177897
  • Chern, Hua-Huai; Hwang, Hsien-Kuei; Yeh, Yeong-Nan. Distribution of the number of consecutive records. Proceedings of the Ninth International Conference "Random Structures and Algorithms'' (Poznan, 1999). Random Structures Algorithms 17 (2000), no. 3-4, 169--196. MR1801131
  • Csörgö, S.; Wu, W. B. On sums of overlapping products of independent Bernoulli random variables. Ukrain. Mat. Zh. 52 (2000), no. 9, 1304--1309; translation in Ukrainian Math. J. 52 (2000), no. 9, 1496--1503 (2001) MR1816943
  • Goncharov, V. (1944). On the field of combinatory analysis. Soviet Math. Izv., Ser. Math., 8, 3-48. In Russian.
  • Hahlin, L.O. (1995). Double Records. Research Report # 12, Department of Mathematics, Uppsala University.
  • Hirano, K.; Aki, S.; Kashiwagi, N.; Kuboki, H. On Ling's binomial and negative binomial distributions of order $k$. Statist. Probab. Lett. 11 (1991), no. 6, 503--509. MR1116744
  • Holst, Lars. The number of two consecutive successes in a Hoppe-Polya urn. J. Appl. Probab. 45 (2008), no. 3, 901--906. MR2455191
  • Holst, Lars. Counts of failure strings in certain Bernoulli sequences. J. Appl. Probab. 44 (2007), no. 3, 824--830. MR2355594
  • Huffer, Fred W.; Sethuraman, Jayaram; Sethuraman, Sunder. A study of counts of Bernoulli strings via conditional Poisson processes. Proc. Amer. Math. Soc. 137 (2009), no. 6, 2125--2134. MR2480294
  • Joffe, Anatole; Marchand, Éric; Perron, François; Popadiuk, Paul. On sums of products of Bernoulli variables and random permutations. J. Theoret. Probab. 17 (2004), no. 1, 285--292. MR2054589
  • Joffe, A., Marchand É., Perron, F., and Popadiuk, P. (2000). On sums of products of Bernoulli variables and random permutations. Research Report # 2686, Centre de Recherches Mathématiques, Montréal, Canada.
  • Kolchin, V.F. (1971). A problem of the allocation of particles in cells and cycles of random permutations. Theory of Probability and its Applications, 16, 74-90.
  • Lee, P. A. A diagonal expansion for the $2$-variate Dirichlet probability density function. SIAM J. Appl. Math. 21 1971 155--165. MR0288805
  • Mori, Tamas F. On the distribution of sums of overlapping products. Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 833--841. MR1876470
  • Sethuraman, Jayaram; Sethuraman, Sunder. On counts of Bernoulli strings and connections to rank orders and random permutations. A festschrift for Herman Rubin, 140--152, IMS Lecture Notes Monogr. Ser., 45, Inst. Math. Statist., Beachwood, OH, 2004. MR2126893


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.