A probabilistic error estimate of quadrature formulas accurate for Haar polynomials

Quadrature formulas possessing the Haar $d$-property (i.e., the formulas that are accurate for Haar functions of groups with the numbers not exceeding a given number $d$) are studied. Previously it was proved that these quadrature formulas have the best order of convergence to zero for the error functional on the classes $S_p$ consisting of the functions with the fast convergent Fourier-Haar series. In this paper we obtain a probabilistic error estimate on the classes $S_p$ for the quadrature formulas possessing the Haar $d$-property. According to this estimate, for a function randomly chosen from $S_p$ the order of convergence to zero for the error functional is better with an arbitrarily high probability than that obtained previously. In 1970s, I.M. Sobol studied the quadrature formulas with nodes that form $\Pi_\tau$ grids; these formulas are also accurate for the Haar functions. This paper generalizes the result obtained by Sobol to the case of arbitrary quadrature formulas possessing the Haar $d$-property.