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

• Authors: Filippo Palombi;Simona Toti
• Venue: Journal of Multivariate Analysis
• Year: 2013

We present arguments in favor of the inequalities var(X"n^2|X@?B"v(@r))@?2@l"nE[X"n^2|X@?B"v(@r)], where X~N"v(0,@L) is a normal vector in v=1 dimensions, with zero mean and covariance matrix @L=diag(@l), and B"v(@r) is a centered v-dimensional Euclidean ball of square radius @r. Such relations lie at the heart of an iterative algorithm, proposed by Palombi et al. (2012) [6] to perform a reconstruction of @L from the covariance matrix of X conditioned to B"v(@r). In the regime of strong truncation, i.e. for @r@?@l"n, the above inequality is easily proved, whereas it becomes harder for @r@?@l"n. Here, we expand both sides in a function series controlled by powers of @l"n/@r and show that the coefficient functions of the series fulfill the inequality order by order if @r is sufficiently large. The intermediate region remains at present an open challenge.