Gaps and interleaving of point processes in sampling from a residual allocation model

Standard

Gaps and interleaving of point processes in sampling from a residual allocation model. / Pitman, Jim; Yakubovich, Yuri.

In: Bernoulli, Vol. 25, No. 4B, 11.2019, p. 3623-3651.

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

BibTeX

@article{e3a9d17b132c457f8bd51f8b14bc75f5,

title = "Gaps and interleaving of point processes in sampling from a residual allocation model",

abstract = "This article presents a limit theorem for the gaps Ĝi:n := X n-i+1:n - X n-i:n between order statistics X 1:n≤ · · · ≤ X n:n of a sample of size n from a random discrete distribution on the positive integers (P 1,P 2, . .) governed by a residual allocation model (also called a Bernoulli sieve) P j := H j j-1 i=1 (1 - Hi ) for a sequence of independent random hazard variables Hi which are identically distributed according to some distribution of H ϵ (0, 1) such that -log(1 - H) has a non-lattice distribution with finite mean μlog. As n→∞the finite dimensional distributions of the gaps Ĝ+ i:n converge to those of limiting gaps Gi which are the numbers of points in a stationary renewal process with i.i.d. spacings -log(1-H j ) between times T i-1 and T i of births in a Yule process, that is T i:=Σ i k=1 ek/k for a sequence of i.i.d. exponential variables ϵ k with mean 1. A consequence is that the mean of Ĝ+ i:n converges to the mean of G i, which is 1/(iμ log). This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve. ",

keywords = "GEM distribution, interleaving of simple point processes, residual allocation model, stars and bars duality, stationary renewal process, Yule process, EMPTY BOXES, BRANCHING-PROCESSES, NUMBER, Stationary renewal process, Residual allocation model, Interleaving of simple point processes, Stars and bars duality",

author = "Jim Pitman and Yuri Yakubovich",

note = "Pitman, Jim; Yakubovich, Yuri. Gaps and interleaving of point processes in sampling from a residual allocation model. Bernoulli 25 (2019), no. 4B, 3623--3651. doi:10.3150/19-BEJ1104. https://projecteuclid.org/euclid.bj/1569398779",

year = "2019",

month = nov,

doi = "10.3150/19-bej1104",

language = "Английский",

volume = "25",

pages = "3623--3651",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "International Statistical Institute",

number = "4B",

}

RIS

TY - JOUR

T1 - Gaps and interleaving of point processes in sampling from a residual allocation model

AU - Pitman, Jim

AU - Yakubovich, Yuri

N1 - Pitman, Jim; Yakubovich, Yuri. Gaps and interleaving of point processes in sampling from a residual allocation model. Bernoulli 25 (2019), no. 4B, 3623--3651. doi:10.3150/19-BEJ1104. https://projecteuclid.org/euclid.bj/1569398779

PY - 2019/11

Y1 - 2019/11

N2 - This article presents a limit theorem for the gaps Ĝi:n := X n-i+1:n - X n-i:n between order statistics X 1:n≤ · · · ≤ X n:n of a sample of size n from a random discrete distribution on the positive integers (P 1,P 2, . .) governed by a residual allocation model (also called a Bernoulli sieve) P j := H j j-1 i=1 (1 - Hi ) for a sequence of independent random hazard variables Hi which are identically distributed according to some distribution of H ϵ (0, 1) such that -log(1 - H) has a non-lattice distribution with finite mean μlog. As n→∞the finite dimensional distributions of the gaps Ĝ+ i:n converge to those of limiting gaps Gi which are the numbers of points in a stationary renewal process with i.i.d. spacings -log(1-H j ) between times T i-1 and T i of births in a Yule process, that is T i:=Σ i k=1 ek/k for a sequence of i.i.d. exponential variables ϵ k with mean 1. A consequence is that the mean of Ĝ+ i:n converges to the mean of G i, which is 1/(iμ log). This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.

AB - This article presents a limit theorem for the gaps Ĝi:n := X n-i+1:n - X n-i:n between order statistics X 1:n≤ · · · ≤ X n:n of a sample of size n from a random discrete distribution on the positive integers (P 1,P 2, . .) governed by a residual allocation model (also called a Bernoulli sieve) P j := H j j-1 i=1 (1 - Hi ) for a sequence of independent random hazard variables Hi which are identically distributed according to some distribution of H ϵ (0, 1) such that -log(1 - H) has a non-lattice distribution with finite mean μlog. As n→∞the finite dimensional distributions of the gaps Ĝ+ i:n converge to those of limiting gaps Gi which are the numbers of points in a stationary renewal process with i.i.d. spacings -log(1-H j ) between times T i-1 and T i of births in a Yule process, that is T i:=Σ i k=1 ek/k for a sequence of i.i.d. exponential variables ϵ k with mean 1. A consequence is that the mean of Ĝ+ i:n converges to the mean of G i, which is 1/(iμ log). This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.

KW - GEM distribution

KW - interleaving of simple point processes

KW - residual allocation model

KW - stars and bars duality

KW - stationary renewal process

KW - Yule process

KW - EMPTY BOXES

KW - BRANCHING-PROCESSES

KW - NUMBER

KW - Stationary renewal process

KW - Residual allocation model

KW - Interleaving of simple point processes

KW - Stars and bars duality

UR - http://www.mendeley.com/research/gaps-interleaving-point-processes-sampling-residual-allocation-model

UR - http://www.scopus.com/inward/record.url?scp=85073009283&partnerID=8YFLogxK

U2 - 10.3150/19-bej1104

DO - 10.3150/19-bej1104

M3 - статья

VL - 25

SP - 3623

EP - 3651

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 4B

ER -

ID: 47509206