The "north pole problem" and random orthogonal matrices

This paper is motivated by the following observation. Take a 3x3 random (Haar distributed) orthogonal matrix [Gamma], and use it to "rotate" the north pole, x0 say, on the unit sphere in R3. This then gives a point u=[Gamma]x0 that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving v=[Gamma]u=[Gamma]2x0. 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=[Gamma]3x0 has higher probability of being closer to the poles ±x0 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 Rp, where p>=3. What are the distributions of U2=x'[Gamma]2x and U3=x'[Gamma]3x? 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 x0=(1,0,...,0)'[set membership, variant]Rp. Call this the "north pole". Then is the first component of the vector [Gamma]kx0. We derive stochastic representations for the exact distributions of U2 and U3 in terms of random variables with known distributions.