| Summary: | <p>We study composition-valued Markov chains that appear naturally in the frame-work of up-down Chinese Restaurant Processes (CRPs), introducing first discrete time models as in a similar way to those introduced by Petrov (2009), and also in continuous time via a Poissonisation technique similar to that used in Pal (2011). </p>
<p>As time evolves in the continuous time model, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (2009). We relate such up-down CRPs to the splitting trees of Lambert (2010) inducing spectrally positive Lévy processes. Conversely, we develop a theorem 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 that connects to work by Forman et al. (2018+) on interval partition diffusions. </p>
<p>Fulman 2009 shows that the framework of commutation relations can be used to study the behaviour of discrete time down-up Markov chains, and he applies this to the unordered up-down (α,θ)-CRP from Pitman 2006. We demonstrate that these techniques still yield valid results when applied to the ordered down-up (α,θ)-CRP of Pitman and Winkel 2009, and generalise a theorem of Fulman 2009 on the explicitform and asymptotics of the maximal separation distance from the case θ= 1 to general the case θ > 0.</p>
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