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 ( X n 2 ∣ X ∈ B v ( ρ ) ) ≤ 2 λ n E [ X n 2 ∣ X ∈ B v ( ρ ) ] , where X ∼ N v ( 0 , Λ ) is a normal vector in v ≥ 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 by Palombi et al. (2012) [6] 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 ≲ λ n , the above inequality is easily proved, whereas it becomes harder for ρ ≫ 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.