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.
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