A Consistent Markov Partition Process Generated from the Paintbox Process

Abstract

We study a family of Markov processes on P (k) , the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process onν , where ν is the distribution of the paintbox based on the probability measure ν on P m , the set of ranked-mass partitions of 1, and (k) ν is the product measure on. We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

The cut-and-paste process

“…Decades ago, population genetics applications motivated the initial study of random partitions and partitionvalued processes [9,11,13]. Somewhat later, Bertoin [2,3] and Pitman [14,6 H. CRANE Table 1 An array of DNA sequences for 3 individuals. From this array, we obtain a sequence in {A, C, G, T } [3] : (AAT, ATT, TTT, CCG, CGG, GGC, AAT, .…”

“…Self-similar cut-and-paste processes. In [6], we introduced a family of cut-and-paste chains, which we now call self-similar homogeneous cutand-paste chains. We showed an instance of these chains in Example 1.7.…”

Convergence rates of Markov chains on spaces of partitions

“…Since the state spaces of interest in our main results are finite, it is natural to use the total variation metric to measure the distance between the law D(X m ) of the chain X at time m ≥ 1 and its stationary distribution π. The total variation distance µ − ν TV between two probability measures µ, ν on a finite or countable set X is defined by (2) µ − ν TV = 1 2 x∈X |µ(x) − ν(x)| = max B⊂X (ν(B) − µ(B)).…”

“…We measure distance to stationarity using the total variation metric (2). Write D(X m ) to denote the distribution of X m .…”

“…Example 5.8 (Self-similar cut-and-paste chains). Self-similar cut-and-paste chains were introduced in [2]. These are EFCP chains for which the paintbox measure Σ = Σ ν is such that if S 1 ∼ Σ then the columns of S 1 are i.i.d.…”