Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

We consider the numerical integration $${\mathrm{I}\mathrm{N}\mathrm{T}}_d(f)={\int }_{{\mathbb{B}}^d}f(x)w_{\mu }(x)dx$$ for the weighted Sobolev classes $B{W}_{p,\mu }^r$ and the weighted Besov classes $B{B}_{\tau }^r(L_{p,\mu })$ in the randomized case setting, where $w_{\mu },\mu \ge 0,$ is the classical Jacobi weight on the ball ${\mathbb{B}}^d$, $1\le p\le \mathrm{\infty }$, $r>(d+2\mu )\mathrm{/}p$, and $0<\tau \le \mathrm{\infty }$. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are $n^{-r\mathrm{/}d-1\mathrm{/}2+(1\mathrm{/}p-1\mathrm{/}2{)}_{+}}$. Compared to the orders $n^{-r\mathrm{/}d}$ of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when $p>1$.