On invariant parametric covariance families (Q1193390)

A family \({\mathcal K}=\{\sum(\xi): \xi\in\Xi\}\) of non-negative definite \(k\times k\) matrices is a parametric covariance family with parametric set \(\Xi\in\mathbb{R}^ l\), where \(l\leq k(k+1)/2\), if \(\xi\to\sum(\xi)\) is one-to-one. This paper provides a characterization of all linear transformations of random vectors that are invariant for a given parametric covariance family. Applications to two one-dimensional covariance families are discussed.