Multivariate Gaussian approximations on Markov chaoses

Campese, Simon; Nourdin, Ivan; Peccati, Giovanni; Poly, Guillaume

doi:10.1214/16-ecp4615

Cited by 10 publications

(15 citation statements)

References 14 publications

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“…We remark that the fourth-moment theorem for case (C) has been established by [1] in the univariate case, which inspired our discussions in Sect. 5 (see also [9]).…”

Section: Theorem 22 Let Us Keep the Same Notation As In Corollary 22 ...mentioning

confidence: 92%

“…Proposition 5.1 in [39]). Moreover, as clarified in [1,9,38], only some spectral properties of the Markov diffusion operator are essential for deriving a fourth-moment-type bound for the variance of the corresponding Stein kernel. This spectral property is especially satisfied when each X i is either a Gaussian or (standardized) gamma variable, so this section focuses on such a situation and derives a high-dimensional CLT for this special case.…”

Section: A High-dimensional Clt For Normal-gamma Homogeneous Sumsmentioning

confidence: 99%

“…When f = g, Azmoodeh et al [1] have derived a better estimate than (5.4) in a more general setting. Their technique of the proof can also be applied to the case f = g, and this has been implemented in Campese et al [9]. However, this leads to a bound containing the quantity Cov[F 2 , G 2 ] − 2 E [FG] 2 , so we need an additional argument to estimate it.…”

Section: A Bound For the Variance Of The Carré Du Champ Operatormentioning

confidence: 99%

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High-Dimensional Central Limit Theorems for Homogeneous Sums

Koike

2022

J Theor Probab

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This paper develops a quantitative version of de Jong’s central limit theorem for homogeneous sums in a high-dimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multi-dimensional vector of homogeneous sums and a Gaussian vector so that the bound depends polynomially on the logarithm of the dimension and is governed by the fourth cumulants and the maximal influences of the components. As a corollary, we obtain high-dimensional versions of fourth-moment theorems, universality results and Peccati–Tudor-type theorems for homogeneous sums. We also sharpen some existing (quantitative) central limit theorems by applications of our result.

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“…We remark that the fourth-moment theorem for case (C) has been established by [1] in the univariate case, which inspired our discussions in Sect. 5 (see also [9]).…”

Section: Theorem 22 Let Us Keep the Same Notation As In Corollary 22 ...mentioning

confidence: 92%

Section: A High-dimensional Clt For Normal-gamma Homogeneous Sumsmentioning

confidence: 99%

Section: A Bound For the Variance Of The Carré Du Champ Operatormentioning

confidence: 99%

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High-Dimensional Central Limit Theorems for Homogeneous Sums

Koike

2022

J Theor Probab

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“…Another important ingredient in our proof is Ledoux spectral point-of-view for fourth moment theorem [11], which was later refined e.g. in [1,2]. Such a spectral viewpoint helps one get rid of some computational deadlock that is usually caused by the complicated multiplication formula.…”

Section: Strategy Of Proving Theorem 11mentioning

confidence: 99%

“…This proves (3.12), while the particular case follows from again the Stroock's formula. More precisely, one can first deduce from the previous discussion that Q 1 (h) 2 2 . By the definition of discrete gradient, one has…”

Section: Proofs Of Technical Lemmasmentioning

confidence: 99%

A Peccati-Tudor type theorem for Rademacher chaoses

Zheng

2019

ESAIM: PS

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In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2017).Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition. Abstract 2010 Mathematics Subject Classification. Primary: 60F05, 60B12; Secondary: 47N30.

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Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

et al. 2016

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The aim of the present paper is to establish the multidimensional counterpart of the fourth moment criterion for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in [22] in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

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Multivariate Gaussian approximations on Markov chaoses

Cited by 10 publications

References 14 publications

High-Dimensional Central Limit Theorems for Homogeneous Sums

High-Dimensional Central Limit Theorems for Homogeneous Sums

A Peccati-Tudor type theorem for Rademacher chaoses

Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

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