Stochastic Bernstein polynomials: uniform convergence in probability with rates

This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics.The results given here have the following attributes, (1) they extendresults of Newey to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend themain results in the literature in a parsimonious fashion.

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On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.12.28 ◽

2021 ◽

pp. 4903-4915

Author(s):

Ali Jassim Muhammad ◽

Asma Jaber

Keyword(s):

Asymptotic Formula ◽

Uniform Convergence ◽

Convergence Theorem ◽

Recurrence Relations ◽

Bernstein Polynomials ◽

Negative Integer ◽

Asymptotic Formulas ◽

Order Moment ◽

Approximation Properties ◽

Bernstein Type

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions , is a non-negative integer

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On Multivariate Bernstein Polynomials

Mathematics ◽

10.3390/math8091397 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1397

Author(s):

Mama Foupouagnigni ◽

Merlin Mouafo Wouodjié

Keyword(s):

Uniform Convergence ◽

Lower Bounds ◽

Order Of Convergence ◽

Bernstein Polynomials ◽

Upper And Lower Bounds ◽

Fixed Sign

In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.

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Conditions for the uniform convergence in probability of wavelet decompositions for stochastic processes from the space $\operatorname{Exp}_{\varphi}(\Omega)$

Theory of Probability and Mathematical Statistics ◽

10.1090/s0094-9000-2011-00812-5 ◽

2010 ◽

Vol 81 ◽

pp. 85-85

Author(s):

Yu. V. Kozachenko ◽

O. V. Polos’mak

Keyword(s):

Stochastic Processes ◽

Uniform Convergence ◽

Convergence In Probability ◽

Wavelet Decompositions

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Approximation of random functions by stochastic Bernstein polynomials in capacity spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2021.02.04 ◽

2021 ◽

Vol 37 (2) ◽

pp. 185-194

Author(s):

SORIN G. GAL ◽

CONSTANTIN P. NICULESCU

Keyword(s):

Uniform Convergence ◽

Quantitative Estimate ◽

Random Function ◽

Bernstein Polynomials ◽

Random Functions ◽

Quantitative Estimates ◽

Univariate Case ◽

Convergence In Capacity

Given a submodular capacity space, we firstly obtain a quantitative estimate for the uniform convergence in the Choquet p-mean, 1\le p<\infty, of the multivariate stochastic Bernstein polynomials associated to a random function. Also, quantitative estimates concerning the uniform convergence in capacity in the univariate case are given.

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