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.

翻译：涉及不变随机向量的计算与不变量理论直接相关（参见例如[1]）。本文沿此思路提出若干简单观察。例如，任何$O(n)$不变量都是随机张量的期望。此外，$O(n)$不变量标准基元素的平均值等于随机Veronese张量的期望，仅相差一个已知标量倍数。