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February 2022 A Ray–Knight representation of up-down Chinese restaurants
Dane Rogers, Matthias Winkel
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Bernoulli 28(1): 689-712 (February 2022). DOI: 10.3150/21-BEJ1364

Abstract

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.

Acknowledgements

Dane Rogers was supported by EPSRC DPhil studentship award 1512540 and by a Merton doctoral completion bursary. We would like to thank Jim Pitman for pointing out some relevant references and Noah Forman, Soumik Pal and Douglas Rizzolo for allowing us to build on unpublished drafts from which several ideas here arose and that explored the special case α=1/2 specifically. We thank Christina Goldschmidt and Loïc Chaumont for valuable feedback on the thesis version of this paper, which also led to improvements here.

Citation

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Dane Rogers. Matthias Winkel. "A Ray–Knight representation of up-down Chinese restaurants." Bernoulli 28 (1) 689 - 712, February 2022. https://doi.org/10.3150/21-BEJ1364

Information

Received: 1 June 2020; Revised: 1 January 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1364

Keywords: Chinese restaurant process , composition , Ray–Knight theorem , Scaling limit , squared Bessel process , Stable process

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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