It was observed in \cite{Al2} that the expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on the unit sphere in $\mathbb{R}^n $ is equal to $1/n$. It is shown below that this is a characteristic property of random unit vectors defined on invariant probability subspaces of irreducible real representations of compact Lie groups.
Random vectors on the spin configuration of a Curie-Weiss model on Erdős-Rényi random graphs
Efficient simulation of density and probability of large deviations of sum of random vectors using saddle point representations
Gaussian approximations for random vectors
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