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

We present arguments in favor of the inequalities var(X2 n |X ∈ Bv(ρ)) ≤ 2λnE[X n |X ∈ Bv(ρ)], where X ∼ Nv(0,Λ) is a normal vector in v ≥ 1 dimensions, with zero mean and covariance matrix Λ = diag(λ), and Bv(ρ) is a centered v–dimensional Euclidean ball of square radius ρ. Such relations lie at the heart of an iterative algorithm, proposed by Palombi et al. (2012) [6] to perform a reconstruction of Λ from the covariance matrix of X conditioned to Bv(ρ). In the regime of strong truncation, i.e. for ρ . λn, the above inequality is easily proved, whereas it becomes harder for ρ λn. Here, we expand both sides in a function series controlled by powers of λn/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.