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Register a new userUp-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits
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We introduce a three-parameter family of up-down ordered Chinese restaurant processes ${rm PCRP}^{(alpha)}(theta_1,theta_2)$, $alphain(0,1)$, $theta_1,theta_2ge 0$, generalising the two-parameter family of Rogers and Winkel. Our main result establishes self-similar diffusion limits, ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolutions generalising existing families of interval partition evolutions. We use the scaling limit approach to extend 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 $alphain(0,1)$ and $theta:=theta_1+theta_2-alphage-alpha$, including for the first time the usual range of $theta>-alpha$ rather than being restricted to $thetage 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.
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Forman et al. (2020+) constructed $(alpha,theta)$-interval partition evolutions for $alphain(0,1)$ and $thetage 0$, in which the total sums of interval lengths (total mass) evolve as squared Bessel processes of dimension $2theta$, where $thetage 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(alpha,theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${rm SSIP}^{(alpha)}(theta_1,theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $theta_1ge 0$ and $theta_2ge 0$. They also have squared Bessel total mass processes of dimension $2theta$, where $theta=theta_1+theta_2-alphage-alpha$ covers emigration as well as immigration. Under the constraint $max{theta_1,theta_2}gealpha$, we prove that an ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $alpha$ and $theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.
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 (2009). We relate such up-down CRPs to the splitting trees of Lambert (2010) inducing spectrally positive L{e}vy processes. Conversely, we develop theorems of Ray-Knight type to recover more general up-down CRPs from the heights of L{e}vy processes with jumps marked by integer-valued paths. We further establish limit theorems for the L{e}vy process and the integer-valued paths to connect to work by Forman et al. (2018+) on interval partition diffusions and hence to some long-standing conjectures.
We construct a stationary Markov process corresponding to the evolution of masses and distances of subtrees along the spine from the root to a branch point in a conjectured stationary, continuum random tree-valued diffusion that was proposed by David Aldous. As a corollary this Markov process induces a recurrent extension, with Dirichlet stationary distribution, of a Wright-Fisher diffusion for which zero is an exit boundary of the coordinate processes. This extends previous work of Pal who argued a Wright-Fisher limit for the three-mass process under the conjectured Aldous diffusion until the disappearance of the branch point. In particular, the construction here yields the first stationary, Markovian projection of the conjectured diffusion. Our construction follows from that of a pair of interval partition-valued diffusions that were previously introduced by the current authors as continuum analogues of down-up chains on ordered Chinese restaurants with parameters (1/2,1/2) and (1/2,0). These two diffusions are given by an underlying Crump-Mode-Jagers branching process, respectively with or without immigration. In particular, we adapt the previous construction to build a continuum analogue of a down-up ordered Chinese restaurant process with the unusual parameters (1/2,-1/2), for which the underlying branching process has emigration.
The two parameter Poisson-Dirichlet distribution $PD(alpha,theta)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingmans Poisson-Dirichlet distribution. The two parameter Dirichlet process $Pi_{alpha,theta, u_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(alpha,theta)$ and $Pi_{alpha,theta, u_0}$. The methods used come from the theory of Dirichlet forms.
Consider a two-type Moran population of size $N$ subject to selection and mutation, which is immersed in a varying environment. The population is susceptible to exceptional changes in the environment, which accentuate the selective advantage of the fit individuals. In this setting, we show that the type-composition in the population is continuous with respect to the environment. This allows us to replace the deterministic environment by a random one, which is driven by a subordinator. Assuming that selection, mutation and the environment are weak in relation to $N$, we show that the type-frequency process, with time speed up by $N$, converges as $Ntoinfty$ to a Wright--Fisher-type SDE with a jump term modeling the effect of the environment. Next, we study the asymptotic behavior of the limiting model in the far future and in the distant past, both in the annealed and in the quenched setting. Our approach builds on the genealogical picture behind the model. The latter is described by means of an extension of the ancestral selection graph (ASG). The formal relation between forward and backward objects is given in the form of a moment duality between the type-frequency process and the line-counting process of a pruned version of the ASG. This relation yields characterizations of the annealed and the quenched moments of the asymptotic type distribution. A more involved pruning of the ASG allows us to obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.
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