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Stochastic Bernstein polynomials: uniform convergence in probability with rates

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Abstract

We introduce stochastic variants of the classical Bernstein polynomials associated with a continuous function f, built up from a general triangular array of random variables. We discuss the uniform convergence in probability of the approximation process that they represent, providing at the same time rates of convergence. In the particular case in which the triangular array of random variables consists of the uniform order statistics, we give a positive answer to a conjectured raised in Wu and Zhou (Adv. Comput. Math. 46, 8, 2020) about an exponential rate of convergence in probability.

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Funding

This work is partially supported by Research Project PGC2018-097621-B-I00. The second author is also supported by Junta de Andalucía Research Group FQM-0178.

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009, Zaragoza, Spain

    José A. Adell

  2. Departamento de Matemáticas, Universidad de Jaén, 23071, Jaén, Spain

    Daniel Cárdenas-Morales

Authors
  1. José A. Adell
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  2. Daniel Cárdenas-Morales
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Correspondence to Daniel Cárdenas-Morales.

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Communicated by: Tomas Sauer

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Adell, J.A., Cárdenas-Morales, D. Stochastic Bernstein polynomials: uniform convergence in probability with rates. Adv Comput Math 46, 16 (2020). https://doi.org/10.1007/s10444-020-09770-6

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  • DOI: https://doi.org/10.1007/s10444-020-09770-6

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