Guy Louchard ; Werner Schachinger ; Mark Daniel Ward - The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis

dmtcs:9293 - Discrete Mathematics & Theoretical Computer Science, October 2, 2023, vol. 25:2 - https://doi.org/10.46298/dmtcs.9293
The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysisArticle

Authors: Guy Louchard ; Werner Schachinger ; Mark Daniel Ward

    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.


    Volume: vol. 25:2
    Section: Combinatorics
    Published on: October 2, 2023
    Accepted on: June 29, 2023
    Submitted on: April 3, 2022
    Keywords: Mathematics - Probability,05A16, 60C05, 60F05

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