The "north pole problem" and random orthogonal matrices

Details The "north pole problem" and random orthogonal matrices The "north pole problem" and random orthogonal matrices

2008-11-17

arxiv.org/abs/0811.2678v1

Mathematics

Statistics Theory

Scientific paper

This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $\Gamma$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=\Gamma x_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=\Gamma u=\Gamma^2 x_0$. Simulations reported in Marzetta et al (2002) suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, morever, that $w=\Gamma^3 x_0$ has higher probability of being closer to the poles $\pm x_0$ than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension $p\ge 3$, 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 $p\ge 3$. What are the distributions of $U_2=x'\Gamma^2 x$ and $U_3=x'\Gamma^3 x$? It is clear by orthogonal invariance that these distribution do not depend on x, so that we can, without loss of generality, take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then $x_0'\Gamma^ k x_0$ is the first component of the vector $\Gamma^k x_0$. We derive stochastic representations for the exact distributions of $U_2$ and $U_3$ in terms of random variables with known distributions.

Affiliated with

Also associated with

No associations

Rating

The "north pole problem" and random orthogonal matrices does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The "north pole problem" and random orthogonal matrices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The "north pole problem" and random orthogonal matrices will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-100189