Some Remarks on Random Vectors and $O(n)$-Invariants

Abstract

Computations involving invariant random vectors are directly related to the theory of invariants (cf. e.g \cite{Weing_1}). Some simple observations along these lines are presented in this paper. We note in particular that sum of elements of the standard basis of $ O(n)$-invariants is equal to the expectation of a random Veronese tensor up to a known scalar multiplier.