Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Blasi, Pierpaolo De; Mena, Ramsés H.; Prünster, Igor

doi:10.1007/s10463-021-00791-6

Cited by 5 publications

(6 citation statements)

References 34 publications

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“…We are only aware of three works dealing with the case α = 0 (more precisely, its specialization ( 9)). Theorem 1 in [6] provides a two-term expansion of EK n , as n → ∞. Corollary 1.6 of the present paper is a multidimensional (not functional) central limit theorem for the small counts borrowed from p. 379-380 in [2].…”

Section: A Short Survey Of Relevant Literaturementioning

confidence: 72%

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Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Iksanov¹,

Kotelnikova²

2022

Preprint

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A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer j, identify the jth generation individuals with the jth generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights.Assume that there are n balls and that the discrete probability distribution responsible for the first generation is Weibull-like. Denote by K

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Section: A Short Survey Of Relevant Literaturementioning

confidence: 72%

“…Second, it is convenient to have the common normalization (c j f j (T )) 1/2 in all our limit theorems. For instance, this enables us to write formula (6), which greatly facilitates an application of the continuous mapping theorem.…”

Section: Limit Theoremsmentioning

confidence: 99%

Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Iksanov¹,

Kotelnikova²

2022

Preprint

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“…Using a sequence of polynomials, Antoniak (1974) obtained the probability of observing exactly 1 ≤ k ≤ n distinct values in the DP sample and derived the probability mass function of D n . A study on the asymptotic behavior of D n in samples of geometric stick-breaking processes is provided by De Blasi et al (2022), while Ishwaran and Zarepour (2000) examined the behavior of D n for H = beta(a, 1). We will investigate the behavior in general, i.e., for any H over (0, 1).…”

Section: The Behavior Of the Number Of Distinct Valuesmentioning

confidence: 99%

“…Each cluster includes all observations with identical values, whereas the values between clusters are distinct. Much attention has been paid to the number of distinct values to control clustering and deal with mixture models (De Blasi et al, 2022;Ghosal and van der Vaart, 2017;Teh, 2017). Examples are the beta two-parameter process (Ishwaran and Zarepour, 2000) and the geometric stick-breaking process (De Blasi et al, 2022).…”

Section: Introductionmentioning

confidence: 99%

On computing the number of distinct values occurring in generalized Dirichlet process samples

Akell,

Sajadi,

Kazemi

2024

Preprint

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This paper studies the generalized Dirichlet process ( $\mathcal{GDP}$ ) with its main properties, including moments of random weights and tail moments. We present the truncated $\mathcal{GDP}\ as a finite mixture distribution and assess the error bounds caused by the truncation. This tactic provides more practicable stick-breaking priors in nonparametric Bayesian settings and facilitates computation. We obtain the joint density of random weights, show that the number of distinct values varies on raising the $\mathcal{GDP}$ samples, and present the impact of the precision parameter on this number. We also show that our results coincide with the Dirichlet process \((\mathcal{DP})$ . MSC Classification: 62E15 , 60C05 , 97K60

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“…A survey of the literature on the infinite occupancy up to 2007 is given in [9]. An incomplete list of very recent contributions includes [3,4,6,7]. Among other things, the authors of [9] discuss applications of the scheme to ecology, database query optimization and literature.…”

Section: Definition Of the Modelmentioning

confidence: 99%

A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

Iksanov,

Kotelnikova

2024

Modern Stochastics: Theory and Applications

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In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.

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Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Cited by 5 publications

References 34 publications

Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

On computing the number of distinct values occurring in generalized Dirichlet process samples

A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

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