Open Access
August 2020 First-order covariance inequalities via Stein’s method
Marie Ernst, Gesine Reinert, Yvik Swan
Bernoulli 26(3): 2051-2081 (August 2020). DOI: 10.3150/19-BEJ1182


We propose probabilistic representations for inverse Stein operators (i.e., solutions to Stein equations) under general conditions; in particular, we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and nonuniform Stein factors (i.e., bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases and expressed in terms of objects familiar within the context of Stein’s method. Applications of the Cauchy–Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed.


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Marie Ernst. Gesine Reinert. Yvik Swan. "First-order covariance inequalities via Stein’s method." Bernoulli 26 (3) 2051 - 2081, August 2020.


Received: 1 May 2019; Revised: 1 September 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193952
MathSciNet: MR4091101
Digital Object Identifier: 10.3150/19-BEJ1182

Keywords: covariance identities , Cramér–Rao inequality , Stein equation , Stein kernel , Variance bounds

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 3 • August 2020
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