An extended Stein-type covariance identity for the Pearson family, with applications to lower variance bounds

For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. Goldstein and Reinert, 2005, J. Theoret. Probab., vol. 18, pp. 237--260). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, that seems to be known only in particular cases (for the Normal see Houdre and Kagan, 1995, J. Theoret. Probab., vol. 8, pp. 23--30; see also Houdre and Perez-Abreu, 1995, Ann. Probab., vol. 23, pp. 400--419, for corresponding results related to the Wiener and Poisson processes). Some applications are also given.