The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis

TL;DR

This paper investigates the asymptotic behavior of the number of distinct adjacent pairs in geometrically distributed words, providing new formulas and conjectures using probabilistic and combinatorial methods.

Contribution

It offers the first comprehensive analysis of all moments and distributional properties of adjacent pairs in geometric words, combining probabilistic and combinatorial techniques.

Findings

Asymptotic variance for different pairs case

Exact formulas for first and second moments

Conjectures on all moments and distributions

Abstract

The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.