A Ray–Knight representation of up-down Chinese restaurants
We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (Ann....
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Bernoulli Society for Mathematical Statistics and Probability
2021
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| _version_ | 1797109275580956672 |
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| author | Rogers, D Winkel, M |
| author_facet | Rogers, D Winkel, M |
| author_sort | Rogers, D |
| collection | OXFORD |
| description | We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (Ann. Probab. 37 (2009) 1999–2041). We relate such up-down CRPs to the splitting trees of Lambert (Ann. Probab. 38 (2010) 348–395) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray–Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establish limit theorems for the Lévy process and the integer-valued paths to connect to work by Forman, Pal, Rizzolo, Shi and Winkel on interval partition diffusions and hence to some long-standing conjectures. |
| first_indexed | 2024-03-07T07:38:09Z |
| format | Journal article |
| id | oxford-uuid:87e0f8ef-ef57-4063-851a-3c0cd3992018 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:38:09Z |
| publishDate | 2021 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| record_format | dspace |
| spelling | oxford-uuid:87e0f8ef-ef57-4063-851a-3c0cd39920182023-04-04T06:23:33ZA Ray–Knight representation of up-down Chinese restaurantsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:87e0f8ef-ef57-4063-851a-3c0cd3992018EnglishSymplectic ElementsBernoulli Society for Mathematical Statistics and Probability2021Rogers, DWinkel, MWe study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (Ann. Probab. 37 (2009) 1999–2041). We relate such up-down CRPs to the splitting trees of Lambert (Ann. Probab. 38 (2010) 348–395) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray–Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establish limit theorems for the Lévy process and the integer-valued paths to connect to work by Forman, Pal, Rizzolo, Shi and Winkel on interval partition diffusions and hence to some long-standing conjectures. |
| spellingShingle | Rogers, D Winkel, M A Ray–Knight representation of up-down Chinese restaurants |
| title | A Ray–Knight representation of up-down Chinese restaurants |
| title_full | A Ray–Knight representation of up-down Chinese restaurants |
| title_fullStr | A Ray–Knight representation of up-down Chinese restaurants |
| title_full_unstemmed | A Ray–Knight representation of up-down Chinese restaurants |
| title_short | A Ray–Knight representation of up-down Chinese restaurants |
| title_sort | ray knight representation of up down chinese restaurants |
| work_keys_str_mv | AT rogersd arayknightrepresentationofupdownchineserestaurants AT winkelm arayknightrepresentationofupdownchineserestaurants AT rogersd rayknightrepresentationofupdownchineserestaurants AT winkelm rayknightrepresentationofupdownchineserestaurants |