Colloquium Speaker - Dr. Éric Marchand, Université de Sherbrooke

Abstract:

I will provide a personalized account of a sequence of problems, that I have worked on over the years, beginning with string counts in Bernoulli sequences and transitioning to multivariate discrete models. As a starting point, we consider independent Bernoulli trials \( X_k \sim \mathrm{Ber}(p_k = \tfrac{1}{k}), \; k \geq 1 \), and the problem of establishing that the distribution of \( S = \lim_{n \to \infty} S_n \), with \( S_n = \sum_{k=1}^{n} X_k X_{k+1} \), is Poisson(1). We will explain how this finding connects to cycles in random permutations, records for continuous random variables, the Hoppe-Pólya urn, and the classical Montmort’s matching problem.

Extensions to \( p_k = \tfrac{a}{a + b + k - 1} \), with \( a > 0, b \geq 0 \), will be discussed, and we present a multivariate version with Bernoulli arrays \( \{ X_{k,j} \sim \mathrm{Ber}(p_{k,j}), \; k \geq 1, j = 1, \ldots, r + 1 \} \) having multinomial independently distributed rows, and the study of the joint distribution of column sums \( S_j = \sum_{k=1}^{\infty} X_{k,j} X_{k+1,j}, \; j = 1, \ldots, r \). For \( p_{k,j} = \tfrac{1}{b + k} \), \( k \geq 1, j = 1, \ldots, r \), and \( b \geq r - 1 \), a multivariate Poisson mixture with Dirichlet mixing arises and relates to multivariate discrete models with common margins, as well as to a Sum and Shares model developed with Chris Jones, to a multivariate splitting model by Peyhardi et al. (2021), and to a tree Pólya splitting model by Valiquette et al. (2025).