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.
"Measures with zeros in the inverse of their moment matrix." Ann. Probab. 36 (4) 1453 - 1471, July 2008. https://doi.org/10.1214/07-AOP365