Discrepancy for randomized Riemann sums

Given a finite sequence $U_{N}=\{u_{1},\ldots,u_{N}\}$ of points contained in the $d$-dimensional unit torus, we consider the $L^{2}$ discrepancy between the integral of a given function and the Riemann sums with respect to translations of $U_{N}$. We show that with positive probability, the $L^{2}$ discrepancy of other sequences close to $U_{N}$ in a certain sense preserves the order of decay of the discrepancy of $U_{N}$. We also study the role of the regularity of the given function.