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

TL;DR

This paper develops a unified Stein operator framework to derive a variety of covariance and variance bounds for arbitrary univariate distributions, applicable to both discrete and continuous cases, with numerous concrete examples.

Contribution

It introduces a minimal formalism for Stein operators that yields new covariance identities and variance bounds, extending classical results to a broader setting.

Findings

Derivation of Klaassen-type variance bounds using Cauchy-Schwarz.

Development of Papathanasiou-type variance expansions of arbitrary order.

Establishment of Olkin-Shepp type covariance bounds via matrix Cauchy-Schwarz.

Abstract

In this paper, 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-type variance bounds. A probabilistic representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp 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.