On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance

TL;DR

This paper develops new theoretical tools for analyzing complex Gaussian random fields and quadratic forms, and applies them to create faster, less conservative tests for independence of multiple random vectors.

Contribution

It introduces a general tail probability estimate for Gaussian quadratic forms, analyzes complex Gaussian fields, and applies these to improve independence testing methods.

Findings

New tail probability bounds for Gaussian quadratic forms

Efficient estimators for moments of Gaussian quadratic forms

Faster, less conservative independence tests

Abstract

The paper contains results in three areas: First we present a general estimate for tail probabilities of Gaussian quadratic forms with known expectation and variance. Thereafter we analyze the distribution of norms of complex Gaussian random fields (with possibly dependent real and complex part) and derive representation results, which allow to find efficient estimators for the moments of the associated Gaussian quadratic form. Finally, we apply these results to sample distance multivariance, which is the test statistic corresponding to distance multivariance -- a recently introduced multivariate dependence measure. The results yield new tests for independence of multiple random vectors. These are less conservative than the classical tests based on a general quadratic form estimate and they are (much) faster than tests based on a resampling approach. As a special case this also improves independence tests based on distance covariance, i.e., tests for independence of two random vectors.