Convex ordering of P\'{o}lya random variables and monotonicity of the error estimate of Bernstein-Stancu operators

Abstract

In the present paper we show that in P\'{o}lya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the absolute value of the error of Bernstein-Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proof relies on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of P\'{o}lya random variables with respect to the replacement parameter.