The “north pole problem” and random orthogonal matrices

This paper is motivated by the following observation. Take a 3×3 random (Haar distributed) orthogonal matrix ΓΓ, and use it to “rotate” the north pole, x0x0 say, on the unit sphere in R3R3. This then gives a point u=Γx0u=Γx0 that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform uu, giving v=Γu=Γ2x0v=Γu=Γ2x0. 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 vv is more likely to be in the northern hemisphere than in the southern hemisphere, and, moreover, that w=Γ3x0w=Γ3x0 has higher probability of being closer to the poles ±x0±x0 than the uniformly distributed point uu. In this paper we prove these results, in the general setting of dimension p≥3p≥3, by deriving the exact distributions of the relevant components of uu and vv. The essential questions answered are the following. Let xx be any fixed point on the unit sphere in RpRp, where p≥3p≥3. What are the distributions of U2=x′Γ2xU2=x′Γ2x and U3=x′Γ3xU3=x′Γ3x? It is clear by orthogonal invariance that these distributions do not depend on xx, so that we can, without loss of generality, take xx to be x0=(1,0,…,0)′∈Rpx0=(1,0,…,0)′∈Rp. Call this the “north pole”. Then x0′Γkx0 is the first component of the vector Γkx0Γkx0. We derive stochastic representations for the exact distributions of U2U2 and U3U3 in terms of random variables with known distributions.