Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866917129515696128 |
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| author | Shi, Quan Winkel, Matthias |
| author_facet | Shi, Quan Winkel, Matthias |
| contents | We establish scaling limit theorems for the up-down ordered Chinese restaurant processes (oCRPs) of Rogers and Winkel as processes in a space of interval partitions. As previously conjectured, the limits are self-similar diffusions previously constructed directly in the continuum. We extend the oCRP model and the results to a three-parameter family ${\rm oCRP}^{(α)}(θ_1,θ_2)$, $α\in(0,1)$, $θ_1,θ_2\ge 0$. We use the scaling limit approach to extend existing stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $α\in(0,1)$ and $θ:=θ_1+θ_2-α\ge-α$, including for the first time the usual range of $θ>-α$ rather than being restricted to $θ\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_15758 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits Shi, Quan Winkel, Matthias Probability 60J80, 60C05, 60F17 We establish scaling limit theorems for the up-down ordered Chinese restaurant processes (oCRPs) of Rogers and Winkel as processes in a space of interval partitions. As previously conjectured, the limits are self-similar diffusions previously constructed directly in the continuum. We extend the oCRP model and the results to a three-parameter family ${\rm oCRP}^{(α)}(θ_1,θ_2)$, $α\in(0,1)$, $θ_1,θ_2\ge 0$. We use the scaling limit approach to extend existing stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $α\in(0,1)$ and $θ:=θ_1+θ_2-α\ge-α$, including for the first time the usual range of $θ>-α$ rather than being restricted to $θ\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range. |
| title | Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits |
| topic | Probability 60J80, 60C05, 60F17 |
| url | https://arxiv.org/abs/2012.15758 |