The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis
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...
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Discrete Mathematics & Theoretical Computer Science
2023-10-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | http://dmtcs.episciences.org/9293/pdf |
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| author | Guy Louchard Werner Schachinger Mark Daniel Ward |
| author_facet | Guy Louchard Werner Schachinger Mark Daniel Ward |
| author_sort | Guy Louchard |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8a6347e4fe1a4b369bd2ecfed0dcdcb5 |
| institution | OA Journals |
| issn | 1365-8050 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj-art-8a6347e4fe1a4b369bd2ecfed0dcdcb52025-08-20T01:49:32ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502023-10-01vol. 25:2Combinatorics10.46298/dmtcs.92939293The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysisGuy LouchardWerner SchachingerMark Daniel WardThe 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.http://dmtcs.episciences.org/9293/pdfmathematics - probability05a16, 60c05, 60f05 |
| spellingShingle | Guy Louchard Werner Schachinger Mark Daniel Ward The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis Discrete Mathematics & Theoretical Computer Science mathematics - probability 05a16, 60c05, 60f05 |
| title | The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis |
| title_full | The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis |
| title_fullStr | The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis |
| title_full_unstemmed | The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis |
| title_short | The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis |
| title_sort | number of distinct adjacent pairs in geometrically distributed words a probabilistic and combinatorial analysis |
| topic | mathematics - probability 05a16, 60c05, 60f05 |
| url | http://dmtcs.episciences.org/9293/pdf |
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