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Register a new userRanked masses in two-parameter Fleming-Viot diffusions
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In previous work, we constructed Fleming--Viot-type measure-valued diffusions (and diffusions on a space of interval partitions of the unit interval $[0,1]$) that are stationary with the Poisson--Dirichlet laws with parameters $alphain(0,1)$ and $thetageq 0$. In this paper, we complete the proof that these processes resolve a conjecture by Feng and Sun (2010) by showing that the processes of ranked atom sizes (or of ranked interval lengths) of these diffusions are members of a two-parameter family of diffusions introduced by Petrov (2009), extending a model by Ethier and Kurtz (1981) in the case $alpha=0$. The latter diffusions are continuum limits of up-down Chinese restaurant processes.
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Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.
We study the Fleming-Viot particle process formed by N interacting continuous-time asymmetric random walks on the cycle graph, with uniform killing. We show that this model has a remarkable exact solvability, despite the fact that it is non-reversible with non-explicit invariant distribution. Our main results include quantitative propagation of chaos and exponential ergodicity with explicit constants, as well as formulas for covariances at equilibrium in terms of the Chebyshev polynomials. We also obtain a bound uniform in time for the convergence of the proportion of particles in each state when the number of particles goes to infinity.
In this paper we give a central limit theorem for the weighted quadratic variations process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations $sum_{i=1}^{[n s]} sum_{j=1}^{[n t]} | Delta_{i,j} Y |^2$ of a two-parameter diffusion $Y=(Y_{(s,t)})_{(s,t)in[0,1]^2}$ observed on a regular grid $G_n$ is an asymptotically normal estimator of the quadratic variation of $Y$ as $n$ goes to infinity.
We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming--Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming--Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identified as a mixture of Polya urns, whereby the observations can be values from the baseline distribution or copies of previous draws collected at the same time as in the usual P`olya urn, or can be sampled from a random subset of the data collected at previous times. We characterise the time-dependent weights of the mixture which select such subsets and discuss the asymptotic regimes. We describe the induced partition by means of a Chinese restaurant process metaphor with a conveyor belt, whereby new customers who do not sit at an occupied table open a new table by picking a dish either from the baseline distribution or from a time-varying offer available on the conveyor belt. We lay out explicit algorithms for exact and approximate posterior sampling of both observations and partitions, and illustrate our results on predictive problems with synthetic and real data.
We study interval partition diffusions with Poisson--Dirichlet$(alpha,theta)$ stationary distribution for parameters $alphain(0,1)$ and $thetage 0$. This extends previous work on the cases $(alpha,0)$ and $(alpha,alpha)$ and builds on our recent work on measure-valued diffusions. We work on spaces of interval partitions with $alpha$-diversity. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. The additional order and diversity structure of such interval partitions is essential for applications to continuum random tree models such as stable CRTs and limit structures of other regenerative tree growth processes, where intervals correspond to masses of spinal subtrees (or spinal bushes) in spinal order and diversities give distances between any two spinal branch points. We further show that our processes can be extended to enter continuously from the Hausdorff completion of our state space and that, in contrast to the measure-valued setting, these extensions are Feller processes.
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