An ergodic theorem for partially exchangeable random partitions

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An ergodic theorem for partially exchangeable random partitions. / Pitman, Jim; Yakubovich, Yuri.

In: Electronic Communications in Probability, Vol. 22, 64, 2017.

Research output: Contribution to journal › Article › peer-review

Author

Pitman, Jim ; Yakubovich, Yuri. / An ergodic theorem for partially exchangeable random partitions. In: Electronic Communications in Probability. 2017 ; Vol. 22.

BibTeX

@article{7bb1a297804d42b7bebf430f40120cb6,

title = "An ergodic theorem for partially exchangeable random partitions",

abstract = "We consider shifts ∏n,m of a partially exchangeable random partition ∏∞ of ℕ obtained by restricting ∏∞ to {n+1, n+2,…, n+m} and then subtracting n from each element to get a partition of [m]:= {1,…,m}. We show that for each fixed m the distribution of ∏n,m converges to the distribution of the restriction to [m] of the exchangeable random partition of N with the same ranked frequencies as ∏∞. As a consequence, the partially exchangeable random partition ∏∞ is exchangeable if and only if ∏∞ is stationary in the sense that for each fixed m the distribution of ∏n,m on partitions of [m] is the same for all n. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.",

keywords = "Ergodic theorem, Exchangeable random partitions, Partially exchangeable random partitions, Shifted partitions, Stationary distribution",

author = "Jim Pitman and Yuri Yakubovich",

year = "2017",

doi = "10.1214/17-ECP95",

language = "English",

volume = "22",

journal = "Electronic Communications in Probability",

issn = "1083-589X",

publisher = "Institute of Mathematical Statistics",

}

RIS

TY - JOUR

T1 - An ergodic theorem for partially exchangeable random partitions

AU - Pitman, Jim

AU - Yakubovich, Yuri

PY - 2017

Y1 - 2017

N2 - We consider shifts ∏n,m of a partially exchangeable random partition ∏∞ of ℕ obtained by restricting ∏∞ to {n+1, n+2,…, n+m} and then subtracting n from each element to get a partition of [m]:= {1,…,m}. We show that for each fixed m the distribution of ∏n,m converges to the distribution of the restriction to [m] of the exchangeable random partition of N with the same ranked frequencies as ∏∞. As a consequence, the partially exchangeable random partition ∏∞ is exchangeable if and only if ∏∞ is stationary in the sense that for each fixed m the distribution of ∏n,m on partitions of [m] is the same for all n. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.

AB - We consider shifts ∏n,m of a partially exchangeable random partition ∏∞ of ℕ obtained by restricting ∏∞ to {n+1, n+2,…, n+m} and then subtracting n from each element to get a partition of [m]:= {1,…,m}. We show that for each fixed m the distribution of ∏n,m converges to the distribution of the restriction to [m] of the exchangeable random partition of N with the same ranked frequencies as ∏∞. As a consequence, the partially exchangeable random partition ∏∞ is exchangeable if and only if ∏∞ is stationary in the sense that for each fixed m the distribution of ∏n,m on partitions of [m] is the same for all n. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.

KW - Ergodic theorem

KW - Exchangeable random partitions

KW - Partially exchangeable random partitions

KW - Shifted partitions

KW - Stationary distribution

UR - http://www.scopus.com/inward/record.url?scp=85038440433&partnerID=8YFLogxK

U2 - 10.1214/17-ECP95

DO - 10.1214/17-ECP95

M3 - Article

AN - SCOPUS:85038440433

VL - 22

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

M1 - 64

ER -

ID: 11511542