Exactly solvable urn models under random replacement schemes and their applications
We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls i...
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| Veröffentlicht in: | Journal of applied probability 2023-09, Vol.60 (3), p.835-854 |
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| container_title | Journal of applied probability |
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| creator | Inoue, Kiyoshi |
| description | We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models. |
| doi_str_mv | 10.1017/jpr.2022.103 |
| format | Article |
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Appl. Probab</addtitle><date>2023-09-01</date><risdate>2023</risdate><volume>60</volume><issue>3</issue><spage>835</spage><epage>854</epage><pages>835-854</pages><issn>0021-9002</issn><eissn>1475-6072</eissn><abstract>We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. 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| ispartof | Journal of applied probability, 2023-09, Vol.60 (3), p.835-854 |
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| subjects | Combinatorial analysis Combinatorics Isomorphism Ordinary differential equations Original Article Partial differential equations Random variables |
| title | Exactly solvable urn models under random replacement schemes and their applications |
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