arXiv subjects

Nicolae R. Pascu

Publications and source records attributed to Nicolae R. Pascu.

5 recordsLinked to original sources

arXiv2022

Convex ordering of Pólya random variables and monotonicity of the error estimate of Bernstein-Stancu operators

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 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 Pólya random variables with respect to the replacement parameter.

math.PR↗

arXiv2018

A Monotonicity Property of a New Bernstein Type Operator

In the present paper we prove that the probabilities of the Pólya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (Pólya urn distribution with no replacement). As a consequence, we show that the random variables with Pólya urn distribution (with negative replacement) are stochastically ordered with respect to the parameter giving the initial distribution of the urn. An equivalent formulation of this result shows that the new Bernstein operator recently introduced in [3] is a monotone operator. The proofs are probabilistic in spirit and rely on various inequalities, some of which are of independent interest (e.g. a refined version of the reversed Cauchy-Bunyakovsky-Schwarz inequality).

math.PR↗

arXiv2018

An improved Popoviciu-type inequality for a new Bernstein-type operator

Recently we introduced a new Bernstein-type operator using Pólya's urn model with negative replacement, and we showed that it satisfies a Popoviciu-type inequality with a constant slightly larger than that of the corresponding inequality for the classical Bernstein operator. In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant in the Popoviciu inequality for the new operator is in fact smaller than the corresponding constant for the Bernstein operator.

math.CA↗

arXiv2017

A new Bernstein-type operator based on Pólya's urn model with negative replacement

Using Pólya's urn model with negative replacement we introduce a new Bernstein-type operator and we show that the new operator improves upon the known estimates for the classical Bernstein operator. We also provide numerical evidence showing that the new operator gives a better approximation when compared to some other classical Bernstein-type operators.

math.CA↗

arXiv2009

Neighborhoods of univalent functions

The main result shows a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighborhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro-Warschawski-Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.

math.CV↗