Infinite-dimensional integration and the multivariate decomposition method

TL;DR

This paper advances the Multivariate Decomposition Method for infinite-dimensional Lebesgue integration, establishing conditions for integral equivalence and introducing a new setting based on bounds of function norms rather than weights.

Contribution

It develops a new framework for MDM that relies on known bounds of function norms, not weights, and explores integral equivalence under convergence conditions.

Findings

Proves Lebesgue integral equals anchored integral under certain conditions.

Introduces a setting where bounds on function norms are used instead of weights.

Provides examples with reproducing kernel Hilbert spaces and unbounded domain integration.

Abstract

We further develop the \emph{Multivariate Decomposition Method} (MDM) for the Lebesgue integration of functions of infinitely many variables $x_1,x_2,x_3,\ldots$ with respect to a corresponding product of a one dimensional probability measure. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the `anchored' integral, independently of the anchor. The MDM assumes that point values of $f_{\mathfrak{u}}$ are available for important subsets ${\mathfrak{u}}$, at some known cost. In this paper we introduce a new setting, in which it is assumed that each $f_{\mathfrak{u}}$ belongs to a normed space $F_{\mathfrak{u}}$, and that bounds $B_{\mathfrak{u}}$ on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ are known. This contrasts with the assumption in many papers that weights $\gamma_{\mathfrak{u}}$, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights $\gamma_{\mathfrak{u}}$ were determined by minimizing an error bound depending on the $B_{\mathfrak{u}}$, the $\gamma_{\mathfrak{u}}$ \emph{and} the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds $B_{\mathfrak{u}}$ are assumed known. We give two examples in which we specialize the MDM: in the first case $F_{\mathfrak{u}}$ is the $|{\mathfrak{u}}|$-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain.