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In this paper we propose a perturbative method for the reconstruction of thecovariance matrix of a multinormal distribution, under the assumption that theonly available information amounts to the covariance matrix of a sphericallytruncated counterpart of the same distribution. We expand the relevantequations up to the fourth perturbative order and discuss the analyticproperties of the first few perturbative terms. We finally compare the proposedapproach with an exact iterative algorithm (presented in Palombi et al. (2017))in the hypothesis that the spherically truncated covariance matrix is estimatedfrom samples of various sizes.

A perturbative approach to the reconstruction of the eigenvalue spectrum of a normal covariance matrix from a spherically truncated counterpart