Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp–type bounds for arbitrary target distributions Marie Ernst ∗ , Gesine Reinert † and Yvik Swan ∗ December 21, 2018 Abstract In this paper, following on from [49, 50, 63] we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use for revisiting the very classical topic of bounding the variance of functionals of random variables. Applying the Cauchy-Schwarz inequality yields first order upper and lower Klaassen [45]-type variance bounds. A probabilistic representation of Lagrange’s identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou [60]-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp [59] type covariance bounds. All results hold for univariate target distribution under very weak assumptions (in particular they hold for continuous and discrete distributions alike). Many concrete illustrations are provided. 1 Introduction Charles Stein’s mathematical legacy is growing at a remarkable pace and many of the techniques and concepts he pioneered are now a staple of contemporary probability theory. The origins of this stream of research lie in two papers: [67], in which the method was first presented in the context of Gaussian approximation, and [20] where the method was first adapted to a non-Gaussian context, namely that of Poisson approximation. As has been noted by many authors since then, the approach can be applied quasi verbatim to any target distribution other than the Gaussian and the Poisson, under the condition that “correct” ad hoc objects be identified which will permit the basic identities to hold. There now exist several excellent books and reviews on Stein’s method and its consequences in various settings, such as [68, 9, 10, 57, 21]. There also exist several non-equivalent general frameworks for the theory covering to large swaths of probability distributions, of which we single out the works [26, 70] for univariate distributions under analytical assumptions, [6, 7] for infinitely divisible distributions and [55, 35, 37] as well as [29] for multivariate densities under diffusive assumptions. A “canonical” differential Stein operator theory is also presented in [49, 50, 63]. Stein’s method can be broken down into a small number of key steps: [A] identification of a (characterizing) linear operator, [B] bounding of solutions to some differential equations related to this operator, [C] probabilistic Taylor expansions and construction of well-designed couplings; see [62] for an overview. Each of these steps has produced an entire ecosystem of “Stein-type objects” (operators, equations, couplings, etc.). These Stein-type objects are in symbiosis with many classical branches of mathematics such as orthogonal polynomials, functional analysis, PDE theory or Markov chain theory and therefore open bridges between Stein’s theory and these important areas of mathematics. More recently, connections with other more contemporary mathematics have been discovered, such as e.g. information theory as in [58, 8], optimal transportation as in [48, 31], and machine learning as in [36, 53, 23]. In the present paper, we pursue the work begun in [49, 50, 63] and adopt a minimal point of view on all the objects concerned, this time concentrating on the solutions to so-called “Stein equations”. ∗ Universit´ e de Li` ege, corresponding author Yvik Swan: yswan@uliege.be. † University of Oxford. 1 arXiv:1812.10344v1 [math.PR] 26 Dec 2018