Approximation of Hilbert-Valued Gaussians on Dirichlet structures

Bourguin, Solesne; Campese, Simon

doi:10.1214/20-ejp551

Cited by 21 publications

(42 citation statements)

References 26 publications

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“…To the best of our expertise, these statements are the first examples of fourth moment theorems for random elements taking value in a metric space without a Hilbert space structure. See [BC20] for related results in a Dirichlet/Hilbert setting.…”

Section: Introduction and (Simplified) Statements Of Main Resultsmentioning

confidence: 99%

The multivariate functional de Jong CLT

Döbler,

Kasprzak,

Peccati

2021

Preprint

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We prove a multivariate functional version of de Jong's CLT (1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on [0, 1] weakly converge in the Skorohod space as soon as their fourth cumulants in t = 1 vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the 'universality of Wiener chaos' phenomenon first observed in Nourdin, .

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Section: Introduction and (Simplified) Statements Of Main Resultsmentioning

confidence: 99%

The multivariate functional de Jong CLT

Döbler,

Kasprzak,

Peccati

2021

Preprint

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“…Stein's method for process approximation is much less developed, with the seminal ideas going back to [Barbour, 1988[Barbour, , 1990. Since then, there has been good progress for Poisson process approximation; e.g., [Chen and Xia, 2004]; but little else outside of some very recent activity, for certain diffusions and Gaussian processes; see [Kasprzak, 2017a,b,c] and [Bourguin and Campese, 2019].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Stein's method for the Poisson-Dirichlet distribution and the Ewens Sampling Formula, with applications to Wright-Fisher models

Gan¹,

Ross²

2019

Preprint

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We provide a general theorem bounding the error in the approximation of a random measure of interest-for example, the empirical population measure of types in a Wright-Fisher model-and a Dirichlet process, which is a measure having Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson-Dirichlet distribution. We apply the result to bound the error in the approximation of the stationary distribution of types in the finite Wright-Fisher model with infinite-alleles mutation structure (not necessarily parent independent) by the Poisson-Dirichlet distribution. An important consequence of our result is an explicit upper bound on the total variation distance between the random partition generated by sampling from a finite Wright-Fisher stationary distribution, and the Ewens Sampling Formula. The bound is small if the sample size n is much smaller than N 1/6 log(N ) −1/2 , where N is the total population size. Our analysis requires a result of separate interest, giving an explicit bound on the second moment of the number of types of a finite Wright-Fisher stationary distribution. The general approximation result follows from a new development of Stein's method for the Dirichlet process, which follows by viewing the Dirichlet process as the stationary distribution of a Fleming-Viot process, and then applying Barbour's generator approach.

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“…A general (and impressive!) contribution on the matter is the recent work by Bourguin and Campese [17], where the authors are able to retrieve several Hilbert space counterparts of the finite-dimensional results discussed in Section 3 above. Bourguin and Campese's approach (whose discussion requires preliminaries that go beyond the scope of our survey) represents a substantial addition to a line of investigation intiated by L. Coutin and L. Decreusefond in the seminal works [26,29,27,28,30].…”

Section: Functional Approximations and Dirichlet Structuresmentioning

confidence: 99%

Malliavin–Stein method: a survey of some recent developments

Azmoodeh¹,

Peccati²,

Yang³

2021

Modern Stochastics: Theory and Applications

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Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin-Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin-Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.

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Approximation of Hilbert-Valued Gaussians on Dirichlet structures

Cited by 21 publications

References 26 publications

The multivariate functional de Jong CLT

The multivariate functional de Jong CLT

Stein's method for the Poisson-Dirichlet distribution and the Ewens Sampling Formula, with applications to Wright-Fisher models

Malliavin–Stein method: a survey of some recent developments

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