The “north pole problem” and random orthogonal matrices

Abstract

This paper is motivated by the following observation. Take a 3×3 random (Haar distributed) orthogonal matrix Γ , and use it to “rotate” the north pole, x 0 say, on the unit sphere in R 3 . This then gives a point u = Γ x 0 that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u , giving v = Γ u = Γ 2 x 0 . Simulations reported in Marzetta et al. [Marzetta, T.L., Hassibi, B., Hochwald, B.M., 2002. Structured unitary space-time autocoding constellations. IEEE Transactions on Information Theory 48 (4) 942–950] suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, moreover, that w = Γ 3 x 0 has higher probability of being closer to the poles ± x 0 than the uniformly distributed point u . In this paper we prove these results, in the general setting of dimension p ≥ 3 , by deriving the exact distributions of the relevant components of u and v . The essential questions answered are the following. Let x be any fixed point on the unit sphere in R p , where p ≥ 3 . What are the distributions of U 2 = x ′ Γ 2 x and U 3 = x ′ Γ 3 x ? It is clear by orthogonal invariance that these distributions do not depend on x , so that we can, without loss of generality, take x to be x 0 = ( 1 , 0 , … , 0 ) ′ ∈ R p . Call this the “north pole”. Then x 0 ′ Γ k x 0 is the first component of the vector Γ k x 0 . We derive stochastic representations for the exact distributions of U 2 and U 3 in terms of random variables with known distributions.