Infinite-Dimensional Monte-Carlo Integration

TL;DR

This paper introduces a novel approach to infinite-dimensional Monte Carlo integration using properties of uniformly distributed sequences, establishing strong law theorems and providing a new proof of Kolmogorov's law.

Contribution

It develops a new method for infinite-dimensional Monte Carlo integration based on uniformly distributed sequences and proves related strong law theorems, offering an alternative proof of Kolmogorov's law.

Findings

Established validity of infinite-dimensional Strong Law type theorems

Developed a new approach for infinite-dimensional Monte Carlo integration

Provided a different proof of Kolmogorov's strong law of large numbers

Abstract

By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in $R^{\infty}$ described in [G.R. Pantsulaia, On uniformly distributed sequences of an increasing family of finite sets in infinite-dimensional rectangles, Real Anal. Exchange. 36 (2) (2010/2011), 325--340 ], a new approach for an infinite-dimensional Monte-Carlo integration is introduced and the validity of some infinite-dimensional Strong Law type theorems are obtained in this paper. In addition, by using properties of uniformly distributed sequences in a unite interval, a new proof of Kolmogorov's strong law of large numbers is obtained which essentially differs from its original proof.