Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits

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 oCRP(α)(θ1,θ2), α∈(0,1), θ1,θ2≥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∈(0,1) and θ:=θ1+θ2−α≥−α, including for the first time the usual range of θ>−α rather than being restricted to θ≥0. This has applications to Fleming–Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.