Azmoodeh, Ehsan ORCID: 0000-0002-0401-793X, Gasbarra, Dario and Gaunt, Robert E
(2023)
An asymptotic approach to proving sufficiency of Stein characterisations.
ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS, 20 (1).
pp. 127-152.
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 Hp the p-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for Hp(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 H3(X) and H4(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: | Article |
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Uncontrolled Keywords: | Stein&rsquo, s method, Stein characterisation, characteristic function, ordinary differential equa-tion, asymptotic analysis, Gaussian polynomial |
Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
Depositing User: | Symplectic Admin |
Date Deposited: | 04 May 2023 09:21 |
Last Modified: | 04 May 2023 09:21 |
DOI: | 10.30757/ALEA.v20-06 |
Open Access URL: | https://arxiv.org/abs/2109.08579 |
Related URLs: | |
URI: | https://livrepository.liverpool.ac.uk/id/eprint/3170147 |