Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions

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.

How a probabilistic analogue of the mean value theorem yields stein-type covariance identities

“…(ii) Representations for the covariance Cov(g(X), w(X)) have recently been given by [17,18] and by [19] in a more general framework by using Stein operators and, among them, a generalized score function. It is worth mentioning that, in our work, the weight functions w(x) and w(x) are related to the score function of X, see (5.2).…”