Existence of Stein kernels under a spectral gap, and discrepancy bounds

Courtade, Thomas A.; Fathi, Max; Pananjady, Ashwin

doi:10.1214/18-aihp898

Cited by 63 publications

(108 citation statements)

References 53 publications

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“…Ord) distributions which are characterized by the fact that their Stein kernel τ p is a second degree polynomial, see Example 3.8. For more on this topic, we also refer to forthcoming [33] as well as [28,36,35] wherein important contributions to the theory of Stein kernels are provided in a multivariate setting.…”

Section: The Operator a Idmentioning

confidence: 99%

First-order covariance inequalities via Stein’s method

Ernst¹,

Reinert²,

Swan³

2020

Bernoulli

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We propose probabilistic representations for inverse Stein operators (i.e. solutions to Stein equations) under general conditions; in particular we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and non-uniform Stein factors (i.e. bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases, and expressed in terms of objects familiar within the context of Stein's method. Applications of the Cauchy-Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed. * Université de Liège, corresponding author Yvik Swan: yswan@uliege.be.

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Section: The Operator a Idmentioning

confidence: 99%

First-order covariance inequalities via Stein’s method

Ernst¹,

Reinert²,

Swan³

2020

Bernoulli

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“…Thus, one retrieves the results of [22]. (iii) Let α ∈ (1, 2) and let µ α be a rotationally invariant α-stable probability measure on R d with Lévy measure defined by…”

Section: Applications To Functional Inequalities For Sd Random Vectorsmentioning

confidence: 73%

“…Rigidity results for infinitely divisible distributions with finite second mo-ment were obtained in [19,Theorem 2.1] whereas the corresponding stability results were obtained in [2,Theorem 4.5] through Stein's method and variational techniques inspired by [22]. Here, for the rotationally invariant α-stable distribution, α ∈ (1, 2), we revisit the method of [22] using the framework of Dirichlet forms. Coupled with a truncation procedure, rigidity and stability of the Poincaré U -functional are stated in Corollary 5.1, Corollary 5.2 and Theorem 5.3.…”

Section: Introductionmentioning

confidence: 99%

On Stein’s method for multivariate self-decomposable laws

Arras¹,

Houdré²

2019

Electron. J. Probab.

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This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with α-stable ones, α ∈ (0, 1]. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincaré-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carré du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincaré-ratio functional of the rotationally invariant α-stable laws, α ∈ (1, 2), are obtained; and as such they recover the classical Gaussian setting as α → 2.

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“…So all that is left to show is (2). As pointed out in [8], this continuity can be proved when a Poincaré inequality holds. Indeed, if we consider an element P = (P 1 , .…”

Section: Construction Via a Free Poincaré Inequalitymentioning

confidence: 83%

“…An interesting immediate corollary is the following reinforcement of Biane's characterization of the semicircular law. It is a free counterpart of a result of [8] on Gaussian distributions. Corollary 3.4.…”

Section: Construction Via a Free Poincaré Inequalitymentioning

confidence: 99%

A note on existence of free Stein kernels

Cébron

Fathi

Mai

2019

Proc. Amer. Math. Soc.

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Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane. MSC: 46L54; 60F05.

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Existence of Stein kernels under a spectral gap, and discrepancy bounds

Cited by 63 publications

References 53 publications

First-order covariance inequalities via Stein’s method

First-order covariance inequalities via Stein’s method

On Stein’s method for multivariate self-decomposable laws

A note on existence of free Stein kernels

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