A Ray–Knight representation of up-down Chinese restaurants

Journal article

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.

Authors: Rogers, D; Winkel, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Journal: Bernoulli
• Volume: 28
• Issue: 1
• Pages: 689-712
• Publication date: 2021-11-10
• Acceptance date: 2021-01-01
• DOI: 10.3150/21-BEJ1364
• ISSN: 1350-7265
• Language: English
• Keywords: FFR