MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX

We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure μ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with μ. 1. Introduction. It is well known that zeros in off-diagonal entries of the inverse M −1 of a n × n covariance matrix M identify pairs of random variables that have no partial correlation (and so are conditionally independent in case of normally distributed vectors); see, for example, Wittaker [7], Corollary 6.3.4. Allowing zeros in the off-diagonal entries of M −1 is particularly useful for Bayesian estimation of regression models in statistics, and is called Bayesian covariance selection. Indeed, estimating a covariance matrix is a difficult problem for large number of variables, and exploiting sparsity in M −1 may yield efficient methods for Graphical Gaussian Models (GGM). For more details, the interested reader is referred to Cripps, Carter and Kohn [3] and the many references therein. The covariance matrix can be thought of as a matrix whose entries are second moments of a measure. This paper focuses on the truncated moment matrices, Md , consisting of moments up to an order determined by d. First, we describe precisely the pattern of zeroes of M −1 d resulting from the measure having a product type structure. Next, we turn to the study of a particular entry of M −1 d being zero. We find that the key is what we call the conditional triangularity property of orthonormal polynomials (OP) up to degree 2d, associated with the measure. To give the flavor of what this means, let, for instance, μ be the joint distribution μ of n random variables X = (X1 ,...,X n) ,a nd let{pσ }⊂ R[X] be its associated family of orthonormal polynomials. When (Xk)k�=i,j is fixed, they can be viewed as polynomials in R[Xi ,X j ]. If in doing so they exhibit a triangular structure [whence,