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

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

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