Random cubatures and quasi-Monte Carlo methods

We establish and examine the deep connection between highly stratified random cubature formulas and quasi-Monte Carlo methods. A class of such formulas, designed to exactly integrate the introduced generalized s-dimensional Haar system, is shown to have additional variance reduction compared to the known theoretical upper bound. We propose several equivalent expressions for the variance within the standard quasi-Monte Carlo setting. The theory of random cubatures is supplemented with both refined versions of known results and completely new facts.