Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees
In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(\alpha...
| Principais autores: | , , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado em: |
2005
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| _version_ | 1826306819282173952 |
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| author | Dong, R Goldschmidt, C Martin, J |
| author_facet | Dong, R Goldschmidt, C Martin, J |
| author_sort | Dong, R |
| collection | OXFORD |
| description | In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$ distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$ distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between $\operatorname {PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated application of the $\mathrm {Frag}_{\alpha}$ operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation--fragmentation duality. |
| first_indexed | 2024-03-07T06:53:40Z |
| format | Journal article |
| id | oxford-uuid:fd64b49f-7c8c-4b1a-8147-6ba56998bba1 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:53:40Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:fd64b49f-7c8c-4b1a-8147-6ba56998bba12022-03-27T13:28:27ZCoagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive treesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fd64b49f-7c8c-4b1a-8147-6ba56998bba1EnglishSymplectic Elements at Oxford2005Dong, RGoldschmidt, CMartin, JIn this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$ distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$ distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between $\operatorname {PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated application of the $\mathrm {Frag}_{\alpha}$ operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation--fragmentation duality. |
| spellingShingle | Dong, R Goldschmidt, C Martin, J Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees |
| title | Coagulation--fragmentation duality, Poisson--Dirichlet distributions and
random recursive trees |
| title_full | Coagulation--fragmentation duality, Poisson--Dirichlet distributions and
random recursive trees |
| title_fullStr | Coagulation--fragmentation duality, Poisson--Dirichlet distributions and
random recursive trees |
| title_full_unstemmed | Coagulation--fragmentation duality, Poisson--Dirichlet distributions and
random recursive trees |
| title_short | Coagulation--fragmentation duality, Poisson--Dirichlet distributions and
random recursive trees |
| title_sort | coagulation fragmentation duality poisson dirichlet distributions and random recursive trees |
| work_keys_str_mv | AT dongr coagulationfragmentationdualitypoissondirichletdistributionsandrandomrecursivetrees AT goldschmidtc coagulationfragmentationdualitypoissondirichletdistributionsandrandomrecursivetrees AT martinj coagulationfragmentationdualitypoissondirichletdistributionsandrandomrecursivetrees |