Abstract
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the part with probability $\beta_s$, according to a law $\sigma$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $\sigma$ is the uniform measure, then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of "analytic" invariant measures. We also derive transience and recurrence criteria for these chains.
Highlights
Introduction and statement of resultsRecall that the Poisson-Dirichlet measure (with parameter 1) can be described as the probability distribution of (Yn)n≥1 on Ω1 obtained by setting Y1 = U1, Yn+1 = Un+1(1 −
Introduction and statement of resultsLet Ω1 denote the space of partitions of 1, that isΩ1 := {p =i≥1 : p1 ≥ p2 ≥ ... ≥ 0, p1 + p2 + . . . = 1} .By size-biased sampling according to a point p ∈ Ω1 we mean picking the j-th part pj with probability pj
It turns out that one may extend the definition of the basic chain to obtain a Poisson-Dirichlet measure with any parameter as an invariant probability measure, generalizing the result of [15]
Summary
Recall that the Poisson-Dirichlet measure (with parameter 1) can be described as the probability distribution of (Yn)n≥1 on Ω1 obtained by setting Y1 = U1, Yn+1 = Un+1(1 −. It turns out that one may extend the definition of the basic chain to obtain a Poisson-Dirichlet measure with any parameter as an invariant probability measure, generalizing the result of [15]. Our first result is the following characterization of those kernels that yield invariant probability measures which are supported on finite (respectively infinite) partitions. To this end, let S := {p ∈ Ω1 | ∃i ≥ 2 : pi = 0} be the set of finite partitions.
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