Two properties of vectors of quadratic forms in Gaussian random variables

TL;DR

This paper investigates the properties and limit behaviors of vectors composed of quadratic forms in Gaussian variables, revealing conditions for their distributional characteristics and convergence.

Contribution

It provides new insights into the distributional properties and limit theorems for vectors of quadratic forms in Gaussian variables, especially under singular distribution conditions.

Findings

Characterization of when such vectors are not absolutely continuous

Conditions for convergence in law of sequences of these vectors

Implications for the structure of distributions of quadratic forms

Abstract

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.