The "north pole problem" and random orthogonal matrices
| dc.creator | Eaton, Morris L. | |
| dc.creator | Muirhead, Robb J. | |
| dc.date | 2008-11-17 | |
| dc.date.accessioned | 2026-07-07T10:18:50Z | |
| dc.date.available | 2026-07-07T10:18:50Z | |
| dc.description | This paper is motivated by the following observation. Take a 3 x 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 $u=Γx_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=Γu=Γ^2 x_0$. 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 $w=Γ^3 x_0$ 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 $U_2=x'Γ^2 x$ and $U_3=x'Γ^3 x$? 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 $x_0'Γ^ k x_0$ 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. | |
| dc.identifier | https://arxiv.org/abs/0811.2678 | |
| dc.identifier | http://arxiv.org/abs/0811.2678 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174323 | |
| dc.subject | Statistics Theory | |
| dc.subject | Probability | |
| dc.subject | Computation | |
| dc.title | The "north pole problem" and random orthogonal matrices | |
| dc.type | text |