A problem of the Allocation of Particles in Cells and Cycles of Random Permutations

Kolchin, V. F.

doi:10.1137/1116005

Cited by 35 publications

(24 citation statements)

References 4 publications

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“…It is due originally to Goncharov [25] and there are now many different proofs. Feller [22] gives a representation of K , as a sum of independent (but not identically distributed) Bernoulli random variables, Shepp and Lloyd [37] use generating functions, Kolchin [29] uses a representation in terms of random allocation of particles into cells. The authors above all considered the case 8 = 1, but their methods extend to general 8.…”

Section: A the Number Of Cyclesmentioning

confidence: 99%

“…Use the result described in Theorem 1, which guarantees the existence of a coupling satisfying C,, 5 Combining these two estimates, we see that Pruuf. First we combine (28) and (29) to conclude that from which it follows that the theorem will be proved if we establish that "…”

Section: The Erdos-turan Lawmentioning

confidence: 99%

“…arc independent random variables, and 3 denotes convergence in distribution. In the case of a uniform random permutation, in which components arc cycles, the Z, are Poisson distributed with mean as shown by Goncharov [25] and Kolchin [29]. In the case of a random mapping function I uniformly chosen from the n" possibilities, the 2, are Poisson distributed with mean as shown by Kolchin [30].…”

Section: Introductionmentioning

confidence: 97%

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Limit Theorems for Combinatorial Structures via Discrete Process Approximations

Arratia

Tavaré

1992

Random Struct Algorithms

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Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article. we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the Erdos-Tur5n Law for the log-order of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of distinct cycle sizes in a random permutation. 0 1992 John Wiley & Sons, Inc.

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Section: A the Number Of Cyclesmentioning

confidence: 99%

Section: The Erdos-turan Lawmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Limit Theorems for Combinatorial Structures via Discrete Process Approximations

Arratia

Tavaré

1992

Random Struct Algorithms

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“…This result had been discovered earlier (see Kolchin (1971) and Hahlin (1995)), but Diaconis's rediscovery sparked a widespread interest in the topic, and very quickly this result was proved and extended by several people using different techniques. Around 1996, Persi Diaconis noticed the surprising fact that, for the Bern(1, 0) sequence, the count Z 1 is Poisson with mean 1!…”

Section: Introductionmentioning

confidence: 65%

Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Huffer

Sethuraman

2012

J. Appl. Probab.

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An infinite sequence (Y 1 , Y 2 , . . . ) of independent Bernoulli random variables with P(Y i = 1) = a/(a + b + i − 1), i = 1, 2, . . . , where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z 1 , Z 2 , Z 3 , . . . of occurrences of patterns or strings of the form {11}, {101}, {1001}, . . . , respectively, in this sequence. The joint distribution of the counts Z 1 , Z 2 , . . . in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1 , Y 2 , . . . , Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1 , the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

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“…In addition, Gibbs [n] (v • , w • ) is called Kolchin's model [31,28], or Gibbs [n] (v • , w • ) has Kolchin's representation [40], which is identified with the collection of terms of random sum X 1 + · · · + X |Π n | conditioned by i X i = n with independent and identically distributed X 1 , X 2 , ..., independent of |Π n |. As we have seen, definition of Gibbs partition depends on contexts, authors, and papers.…”

Section: Introductionmentioning

confidence: 99%

Extreme sizes in Gibbs-type exchangeable random partitions

Mano

2015

Ann Inst Stat Math

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Gibbs-type exchangeable random partitions, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially, Ewens-Pitman partition, which is the sample from the Poisson-Dirichlet process and has been discussed from rather model-specific viewpoints, and a random partition which was recently introduced by Gnedin, are discussed in the details. As by-products, some formulas for the associated partial Bell polynomials are presented.

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A problem of the Allocation of Particles in Cells and Cycles of Random Permutations

Cited by 35 publications

References 4 publications

Limit Theorems for Combinatorial Structures via Discrete Process Approximations

Limit Theorems for Combinatorial Structures via Discrete Process Approximations

Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Extreme sizes in Gibbs-type exchangeable random partitions

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