First-order covariance inequalities via Stein’s method

Ernst, Marie; Reinert, Gesine; Swan, Yvik

doi:10.3150/19-bej1182

“…(We define 0 0 := 1, but this is irrelevant because the bound (4.51) is greater than 1 in this case.) These bounds were obtained using a recent technique of [10] for bounding distances between distributions that builds upon the formalism of [9] for new representations of solutions to Stein equations. For another recent approach to bounding distances between distributions, see [8].…”

Section: )mentioning

confidence: 99%

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Gaunt

¹

2018

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We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [37] for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and it first two derivatives, of the Rayleigh Stein equation.where the test function h is real-valued. The second step is to solve (1.2) for f h and obtain suitable bounds for the solution. Finally, to approximate the distribution of a random variable of interest W by the target distribution q, one may evaluate both sides of (1.2) at W , take expectations and finally take the supremum of both sides over a class of functions H to obtain d H (W, Y ) := sup h∈H |E[h(W )] − E[h(Y )]| = sup h∈H E[Af h (W )].

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“…We begin this section with some generalities concerning the version of Stein's method developed in [6,5]. We only give a summary here and refer the reader unfamiliar with Stein's method to the Supplementary material for more information and background.…”

Section: Stein's Methods For Approximating Radial Distributionsmentioning

confidence: 99%

“…We also denote F (0) (p) the collection of all mean 0 functions under p. Following [5], to p we associate the Stein operators…”

Section: A1 Preliminary Remarks On Stein Operatorsmentioning

confidence: 99%

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Stein’s method and approximating the quantum harmonic oscillator

McKeague¹,

Peköz²,

Swan³

2019

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Hall et al. (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical “worlds.” A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. (2014) and proven by McKeague and Levin (2016) using Stein’s method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein’s method These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual “density approach” Stein solution (see Chatterjee and Shao (2011)) has a singularity.

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“…(We define 0 0 := 1, but this is irrelevant because the bound (4.53) is greater than 1 in this case.) These bounds were obtained using a recent technique of [11] for bounding distances between distributions that builds upon the formalism of [10] for new representations of solutions to Stein equations. For another recent approach to bounding distances between distributions, see [9].…”

Section: )mentioning

confidence: 99%

New error bounds for Laplace approximationviaStein’s method

Gaunt

¹

2021

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We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren \cite {pike} for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

show abstract

First-order covariance inequalities via Stein’s method

Cited by 17 publications

References 79 publications

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Stein’s method and approximating the quantum harmonic oscillator

New error bounds for Laplace approximationviaStein’s method

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