Some new Stein operators for product distributions

Gaunt, Robert E.; Mijoule, Guillaume; Swan, Yvik

doi:10.1214/19-bjps460

Cited by 8 publications

(15 citation statements)

References 23 publications

(42 reference statements)

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“…This is exactly the plan carried out in [8] for variance-gamma distributed random variables, and their approach rests on the preliminary work of [9] who provides unified bounds on the solutions to the variance-gamma Stein equations. Aside from the variance-gamma case discussed in [12,9], there are several other recent references where versions of (1.4) and (1.8) are proposed for complicated probability distributions such as the Kummer-U distribution [27], or the distribution of products of independent random variables [11,10,13]. The common trait of all these is that the resulting identities all involve second or higher order derivatives of the test functions.…”

Section: Introductionmentioning

confidence: 99%

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

Arras

Azmoodeh

Poly

et al. 2019

Stochastic Processes and their Applications

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We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of ℓ 2 (IN ⋆ ), the space of real valued sequences with finite ℓ 2 norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order U -statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.

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Section: Introductionmentioning

confidence: 99%

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

Arras

Azmoodeh

Poly

et al. 2019

Stochastic Processes and their Applications

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“…All these Stein operators are already known to characterise the distribution through several other approaches. Additionally, the Stein operators of [23] for the product of two independent Gaussian random variables with possibly non-zero means, and the Stein operators of [20] for the noncentral chi-square distribution and the distribution of aX 2 + bX + c, a, b, c ∈ R and X ∼ N (0, 1), have linear coefficients and thus characterise the distribution, a fact that was not noted in these works.…”

Section: Description Of the Approach And First Resultsmentioning

confidence: 99%

An asymptotic approach to proving sufficiency of Stein characterisations

Azmoodeh¹,

Gasbarra²,

Gaunt³

2021

Preprint

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In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For X denoting a standard Gaussian random variable and H p the p-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for H p (X), p = 3, 4, . . . , 8, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of p = 3, 4, . . . , 8 independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases p = 1, 2 are already known to be characterising). We leverage our Stein characterisations of H 3 (X) and H 4 (X) to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method.

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“…By the uniqueness of boundary value problems for first order homogeneous linear ODEs, the unique solution to (4.3) is given by φ Y (t) = φ W (t), and so W is equal in law to Y . , gamma [14,39], variance-gamma [23] and McKay Type I distributions [2], as well as the product of two independent Gaussians [24,28], and linear combinations of gamma random variables [2]. All of these Stein operators are already known to characterise the distribution through several other approaches.…”

Section: Stein Characterisations Of H P (X)mentioning

confidence: 99%

“…This is a consequence of the fact that the densities of these distributions satisfy second order differential equations with polynomial coefficients. In recent years, a number of techniques have been developed for obtaining Stein operators in increasingly complex settings, such as the iterated conditioning argument for deriving Stein operators for products of a quite general class of distributions [24,25,27,28] and the Fourier/Malliavin calculus approach used to obtain Stein operators for linear combinations of gamma random variables [1,2].…”

Section: Introductionmentioning

confidence: 99%

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On algebraic Stein operators for Gaussian polynomials

Azmoodeh¹,

Gasbarra²,

Gaunt³

2019

Preprint

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The first essential ingredient to build up Stein's method for a continuous target distribution is to identify a so-called Stein operator, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic Stein operators (see Definition 3.2), and provide a novel algebraic method to find all the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form Y = h(X), where X = (X 1 , . . . , X d ) has i.i.d. standard Gaussian components and h ∈ K[X] is a polynomial with coefficients in the ring K. Our approach links the existence of an algebraic Stein operator with null controllability of a certain linear discrete system. A MATLAB code checks the null controllability up to a given finite time T (the order of the differential operator), and provides all null control sequences (polynomial coefficients of the differential operator) up to a given maximum degree m. This is the first paper that connects Stein's method with computational algebra to find Stein operators for highly complex probability distributions, such as H 20 (X 1 ), where H p is the p-th Hermite polynomial. A number of examples of Stein operators for H p (X 1 ), p = 3, 4, 5, 6, 7, 8, 10, 12, are gathered in the extended Appendix of this arXiv version. We also introduce a widely applicable approach to proving that Stein operators characterise the target distribution, and use it to prove, amongst other examples, that the Stein operators for H p (X 1 ), p = 3, . . . , 8, with minimum possible maximal polynomial degree m characterise their target distribution.

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Some new Stein operators for product distributions

Cited by 8 publications

References 23 publications

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

An asymptotic approach to proving sufficiency of Stein characterisations

On algebraic Stein operators for Gaussian polynomials

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