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(Xn2∣X∈Bv(ρ))≤2λnE[Xn2∣X∈Bv(ρ)], where X∼Nv(0,Λ)X∼Nv(0,Λ) is a normal vector in v≥1v≥1 dimensions, with zero mean and covariance matrix Λ=diag(λ), and Bv(ρ)Bv(ρ) is a centered vv-dimensional Euclidean ball of square radius ρρ. Such relations lie at the heart of an iterative algorithm, proposed by Palombi et al. (2012) [6] to perform a reconstruction of ΛΛ from the covariance matrix of XX conditioned to Bv(ρ)Bv(ρ). In the regime of strong truncation, i.e.   for ρ≲λnρ≲λn, the above inequality is easily proved, whereas it becomes harder for ρ≫λnρ≫λn. Here, we expand both sides in a function series controlled by powers of λn/ρλn/ρ and show that the coefficient functions of the series fulfill the inequality order by order if ρρ is sufficiently large. The intermediate region remains at present an open challenge.