On Probabilistic Convergence Rates of Symmetric Stochastic Bernstein Polynomials

Shenggang Zhang, Qinjiao Gao () and Chungang Zhu
Additional contact information
Shenggang Zhang: School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
Qinjiao Gao: School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
Chungang Zhu: School of Mathematical Sciences, Dalian University of Technology, Dalian 116016, China

Mathematics, 2025, vol. 13, issue 20, 1-15

Abstract: This paper analyzes the exponential convergence properties of Symmetric Stochastic Bernstein Polynomials (SSBPs), a novel approximation framework that combines the deterministic precision of classical Bernstein polynomials (BPs) with the adaptive node flexibility of Stochastic Bernstein Polynomials (SBPs). Through innovative applications of order statistics concentration inequalities and modulus of smoothness analysis, we derive the first probabilistic convergence rates for SSBPs across all L p ( 1 ≤ p ≤ ∞ ) norms and in pointwise approximation. Numerical experiments demonstrate dual advantages: (1) SSBPs achieve comparable L ∞ errors to BPs in approximating fundamental stochastic functions (uniform distribution and normal density), while significantly outperforming SBPs; (2) empirical convergence curves validate exponential decay of approximation errors. These results position SSBPs as a principal solution for stochastic approximation problems requiring both mathematical rigor and computational adaptability.

Keywords: concentration inequality; modulus of continuity; order statistics; stochastic quasi-interpolation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/20/3281/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/20/3281/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:20:p:3281-:d:1770822

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().