An asymptotic approach to proving sufficiency of Stein characterisations

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-09-17 DOI:10.30757/alea.v20-06
E. Azmoodeh, Dario Gasbarra, Robert E. Gaunt
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引用次数: 2

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.
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证明Stein特征充分性的渐近方法
在将Stein的方法扩展到新的目标分布时,第一步是找到一个合适地表征目标分布的Stein算子。在本文中,我们引入了一种广泛适用的技术来证明这些Stein特征的充分性,它可以应用于当Stein算子是多项式系数的线性微分算子时。该方法包括执行渐近分析来证明只有一个特征函数满足与Stein表征相关的某个微分方程。我们使用这种方法证明了所有具有线性系数的Stein算子都表征了它们的目标分布,并在逐例的基础上验证了文献中系数最多为2度的所有多项式Stein算子都是表征的。对于$X$表示标准高斯随机变量,$H_p$表示$p$- Hermite多项式,我们还证明了$H_p(X)$, $p=3,4, $ ldots,8$,具有最小可能度系数的Stein算子表征了它们的目标分布。并且对于$p=3,4,\ldots,8$独立标准高斯随机变量的乘积的Stein算子是表征的(在这两种情况下,$p=1,2$的Stein算子已经知道是表征的)。我们利用我们的$H_3(X)$和$H_4(X)$的Stein特征,从Malliavin演算中推导出这些目标分布的迭代Gamma算子的特征,这在Malliavin-Stein方法的上下文中是自然的。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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