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In this paper, we present a minimal formalism for Stein operators which leadsto different probabilistic representations of solutions to Stein equations.These in turn provide a wide family of Stein-Covariance identities which we putto use for revisiting the very classical topic of bounding the variance offunctionals of random variables. Applying the Cauchy-Schwarz inequality yieldsfirst order upper and lower Klaassen-type variance bounds. A probabilisticrepresentation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder)leads to Papathanasiou-type variance expansions of arbitrary order. A matrixCauchy-Schwarz inequality leads to Olkin-Shepp type covariance bounds. Allresults hold for univariate target distribution under very weak assumptions (inparticular they hold for continuous and discrete distributions alike). Manyconcrete illustrations are provided.

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