An asymptotic approach to proving sufficiency of Stein characterisations



Azmoodeh, Ehsan ORCID: 0000-0002-0401-793X, Gasbarra, Dario and Gaunt, Robert E
(2021) An asymptotic approach to proving sufficiency of Stein characterisations. [Preprint]

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Abstract

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 $H_p(X)$, $p=3,4,\ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,\ldots,8$ 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.

Item Type: Preprint
Additional Information: 26 pages. This is an expansion of Sections 4.3 and 4.4 of arXiv:1912.04605, with new results added
Uncontrolled Keywords: math.PR, math.PR, math.CA, Primary 34A12, 34E05, 60E10, 62E10 Secondary 60F05, 60H07
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 20 Sep 2021 09:03
Last Modified: 15 Mar 2024 17:02
DOI: 10.48550/arxiv.2109.08579
Open Access URL: https://arxiv.org/abs/2109.08579
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3137680