Construction of the beta distributions using the random permutation divisors
A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing tra...
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Vilnius University Press
2024-01-01
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| Series: | Nonlinear Analysis |
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| Online Access: | https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/34009 |
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| author | Gintautas Bareikis Eugenijus Manstavičius |
| author_facet | Gintautas Bareikis Eugenijus Manstavičius |
| author_sort | Gintautas Bareikis |
| collection | DOAJ |
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A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order.
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| format | Article |
| id | doaj-art-332727dafd7e4a199d65c44014c0adbc |
| institution | OA Journals |
| issn | 1392-5113 2335-8963 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj-art-332727dafd7e4a199d65c44014c0adbc2025-08-20T02:14:50ZengVilnius University PressNonlinear Analysis1392-51132335-89632024-01-0129210.15388/namc.2024.29.34009Construction of the beta distributions using the random permutation divisorsGintautas Bareikis0Eugenijus Manstavičius1https://orcid.org/0000-0002-7185-2708Vilnius UniversityVilnius University, A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order. https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/34009random permutationmultiplicative functionEwens distributionquasihypergeometric distributionarcsine law |
| spellingShingle | Gintautas Bareikis Eugenijus Manstavičius Construction of the beta distributions using the random permutation divisors Nonlinear Analysis random permutation multiplicative function Ewens distribution quasihypergeometric distribution arcsine law |
| title | Construction of the beta distributions using the random permutation divisors |
| title_full | Construction of the beta distributions using the random permutation divisors |
| title_fullStr | Construction of the beta distributions using the random permutation divisors |
| title_full_unstemmed | Construction of the beta distributions using the random permutation divisors |
| title_short | Construction of the beta distributions using the random permutation divisors |
| title_sort | construction of the beta distributions using the random permutation divisors |
| topic | random permutation multiplicative function Ewens distribution quasihypergeometric distribution arcsine law |
| url | https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/34009 |
| work_keys_str_mv | AT gintautasbareikis constructionofthebetadistributionsusingtherandompermutationdivisors AT eugenijusmanstavicius constructionofthebetadistributionsusingtherandompermutationdivisors |