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

TL;DR

This paper demonstrates convex ordering of Pólya's urn variables with respect to the replacement parameter and shows that the error estimate of Bernstein-Stancu operators increases monotonically with this parameter.

Contribution

It establishes convex ordering in Pólya's urn model and links this to the monotonicity of Bernstein-Stancu operator error estimates.

Findings

Convex ordering of Pólya random variables with respect to the replacement parameter.

Monotonicity of the error estimate of Bernstein-Stancu operators in convex functions.

Introduction of an interlacing lemma and analysis of first moments of Pólya variables.

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.