Connections Between Bernoulli Strings and Random Permutations

Sethuraman, Jayaram; Sethuraman, Sunder

doi:10.1007/978-1-4419-6263-8_24

Cited by 2 publications

(6 citation statements)

References 12 publications

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“…However, as an illustration, we focus in two particular cases and compare their behaviours. The results are obtained by using techniques from recent works in the study of pattern strings in Bernoulli sequences (see, for instance [13,14,15,20]).…”

Section: Characterization Of Switch Sequencementioning

confidence: 99%

“…We consider the assumptions i.i.d. and the collection λ i = a/(a + b + i − 1) for a > 0, b ≥ 0 and i ≥ 1, usually denoted by Bern(a, b) (see [14,20]).…”

Section: Example 7 (Relation Between Sq and Example 2)mentioning

confidence: 99%

“…For instance, the sequence Bern(1, 0) arises in the limit in the study of cycles in random permutations and record values of continuous random variables. Moreover, the sequence Bern(a, 0) has some applications in nonparametric Bayesian inference and species allocation models (see [14,20] and references therein).…”

Section: Propositionmentioning

confidence: 99%

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Non-Markovian random walks with memory lapses

González-Navarrete

Lambert

2018

Journal of Mathematical Physics

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We propose an approach to construct Bernoulli trials {X i , i ≥ 1} combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past S i = X 1 + . . . + X i , defines a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials {Y i , i ≥ 1}, called memory switch sequence. We introduce the concept of memory lapses property, which is characterized by intervals of consecutive independent steps in BSRD. The main results include classical limit theorems for a class of linear BSRD. In particular, we obtain a central limit theorem for a class of BSRD which generalizes some previous results in literature. Along the paper, several examples of potential applications are provided.

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Section: Characterization Of Switch Sequencementioning

confidence: 99%

“…We consider the assumptions i.i.d. and the collection λ i = a/(a + b + i − 1) for a > 0, b ≥ 0 and i ≥ 1, usually denoted by Bern(a, b) (see [14,20]).…”

Section: Example 7 (Relation Between Sq and Example 2)mentioning

confidence: 99%

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Non-Markovian random walks with memory lapses

González-Navarrete

Lambert

2018

Journal of Mathematical Physics

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Section: Introductionmentioning

confidence: 99%

“…for every fixed k, as well as estimates of total variation error in this approximation which are useful for k = o(n): see [1, Theorems 1 and 3]. See also Sethuraman and Sethuraman [15] for a review of studies of the distribution of the numbers of ℓ-spacings in infinite sequences of independent Bernoulli trials with sequences of probabilities p i other than the sequence p i = θ/(θ + i − 1) involved in this coupling with a sequence of Ewens(θ) permutations. The coupling described above provides a sequence of Ewens(θ) random permutations (π n,θ ) n≥1 whose cycle structures for different values of n are strongly related: from π n,θ to π n+1,θ , either a single fixed point is added, or a single cycle of π n,θ has its length increased by one.…”

Section: Introductionmentioning

confidence: 99%

Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials

Najnudel¹,

Pitman²

2020

Electron. Commun. Probab.

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Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of [n] and the spacings between successes in a sequence of n independent Bernoulli trials with success probability 1/n at the nth trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens (θ) distribution with spacings derived from independent Bernoulli trials with success probability θ/(n−1+θ) at the nth trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length are independent Poisson variables with means θ/ . Ignatov (1978) first discovered this remarkable result in the uniform case θ = 1, by constructing Bernoulli (1/n) trials as the indicators of record values in a sequence of i.i.d. uniform [0, 1] variables.In the present article, the Poisson property of inhomogeneous Bernoulli spacings is explained by a variation of Ignatov's approach for a general θ > 0. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

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Connections Between Bernoulli Strings and Random Permutations

Cited by 2 publications

References 12 publications

Non-Markovian random walks with memory lapses

Non-Markovian random walks with memory lapses

Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials

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