On conditional moments of high-dimensional random vectors given lower-dimensional projections

TL;DR

This paper demonstrates that many non-Gaussian high-dimensional distributions exhibit approximately linear conditional means and constant conditional variances, extending classical Gaussian properties to broader distribution classes with explicit non-asymptotic results.

Contribution

It extends Gaussian conditional moment properties to a wide class of non-Gaussian distributions in high dimensions with explicit bounds and multiple conditioning variables.

Findings

Conditional moments are approximately Gaussian-like in high dimensions.

Explicit non-asymptotic bounds are provided.

Results generalize earlier Gaussian-specific findings.

Abstract

One of the most widely used properties of the multivariate Gaussian distribution, besides its tail behavior, is the fact that conditional means are linear and that conditional variances are constant. We here show that this property is also shared, in an approximate sense, by a large class of non-Gaussian distributions. We allow for several conditioning variables and we provide explicit non-asymptotic results, whereby we extend earlier findings of Hall and Li (1993) and Leeb (2013).