Abstract
In the present paper we show that in Pólya’s urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a natural convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the error of approximation for Bernstein–Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proofs rely on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of Pólya random variables with respect to the replacement parameter.
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Meleşteu, A.D., Pascu, M.N. & Pascu, N.R. Convex Ordering of Pólya Random Variables and Approximation Monotonicity of Bernstein–Stancu Operators. Results Math 78, 32 (2023). https://doi.org/10.1007/s00025-022-01802-5
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DOI: https://doi.org/10.1007/s00025-022-01802-5
Keywords
- Convex ordering
- Pólya urn model
- Bernstein operator
- error estimate of Bernstein–Stancu operators
- approximation theory