On Probabilistic Convergence Rates of Symmetric Stochastic Bernstein Polynomials

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 Lp (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.