A note on the variance of the square components of a normal multivariate within a Euclidean ball

TL;DR

This paper investigates inequalities related to the variance of squared components of a multivariate normal vector within a Euclidean ball, providing series expansions to support these inequalities in certain regimes.

Contribution

It introduces a series expansion approach to analyze variance inequalities for truncated multivariate normal vectors, extending understanding in the regime of strong truncation.

Findings

Inequalities hold easily for strong truncation regimes.

Series expansions confirm inequalities for large radii.

Open problem remains for intermediate truncation regimes.

Abstract

We present arguments in favour of the inequalities $var(X_n^2|X \in B_v(\rho)) \le 2\lambda_n E[X_n^2|X \in B_v(\rho)]$, where $X \sim N_v(0,\Lambda)$ is a normal vector in $v\ge 1$ dimensions, with zero mean and covariance matrix $\Lambda = \diag(\lambda)$, and $B_v(\rho)$ is a centered $v$-dimensional Euclidean ball of square radius $\rho$. Such relations lie at the heart of an iterative algorithm, proposed in ref. [1] to perform a reconstruction of $\Lambda$ from the covariance matrix of $X$ conditioned to $B_v(\rho)$. In the regime of strong truncation, i.e. for $\rho \lesssim \lambda_n$, the above inequality is easily proved, whereas it becomes harder for $\rho \gg \lambda_n$. Here, we expand both sides in a function series controlled by powers of $\lambda_n/\rho$ and show that the coefficient functions of the series fulfill the inequality order by order if $\rho$ is sufficiently large. The intermediate region remains at present an open challenge.