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

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 $Λ= \diag(λ)$, 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.

26 pages, 6 figures. Added refs. [2] and [7], one paragraph at the end of sect. 1 and fig. 5. Results unchanged

References in corpus (1)

• A perturbative approach to the reconstruction of the eigenvalue spectrum of a normal covariance matrix from a spherically truncated counterpart

Cited by in corpus (4)

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