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

Computations involving invariant random vectors are directly related to the theory of invariants (cf. e.g [1]). Some simple observations along these lines are presented in this paper. For example, any $O(n)$-invariant is an expectation of a random tensor. Moreover, the average 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.