Numerical Integration at the Intersection of Probability Theory, Number Theory and Dynamical Systems

Abstract

This paper is devoted to the study of uniformly distributed (u.d.) sequences on general non-compact topological spaces. Convergence-determining classes and the equivalent characterizations of u.d. sequences on $T_4$ spaces are discussed. For Polish spaces endowed with regular probability measures, the existence of a countable convergence-determining class for u.d. sequences is proved by virtue of the tightness of Borel probability measures. It is shown that, under the condition that a countable convergence-determining class exists, both independent identically distributed (i.i.d.) randomly sampled sequences and orbit sequences of ergodic transformations are u.d.\ with full measure.