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

## TL;DR

This paper explores the relationship between invariant random vectors and the theory of invariants, highlighting a specific connection involving the sum of basis elements and the expectation of a Veronese tensor.

## Contribution

It presents new observations linking invariant random vectors to classical invariant theory, particularly relating sums of basis elements to expectations of Veronese tensors.

## Key findings

- Sum of basis elements equals expectation of a Veronese tensor scaled by a known factor
- Provides insights into the structure of $O(n)$-invariants
- Connects invariant theory with probabilistic computations

## 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.