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
    Funding:
      Source : OpenAIRE Graph
    • Emerging Frontiers of Science of Information; Funder: National Science Foundation; Code: 0939370
    • NSF Convergence Accelerator Track H: Developing Experiential Accessible Framework for Partnerships and Opportunities in Data Science (for the deaf community); Funder: National Science Foundation; Code: 2235473
    • HDR DSC: National Data Mine Network; Funder: National Science Foundation; Code: 2123321
    • MCTP: Sophomore Transitions: Bridges into a Statistics Major and Big Data Research Experiences via Learning Communities; Funder: National Science Foundation; Code: 1246818
    • Category I: Anvil - A National Composable Advanced Computational Resource for the Future of Science and Engineering; Funder: National Science Foundation; Code: 2005632
    • HDR Institute: Geospatial Understanding through an Integrative Discovery Environment; Funder: National Science Foundation; Code: 2118329

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