# order statistics for components of a random unit vector

Suppose you sample uniformly from the unit vectors in R^n. What are the distributions of the order statistics of the magnitudes of the components of the sampled vectors? That is, for 1 <= i <= n and x in [0,1], what is the probability that the i'th largest component of the vector (in absolute value) is less than or equal to x?

• It doesn't answer your question, but would it help to model the uniform distribution on the sphere by a random vector $(X_1/R, X_2/R, ..., X_n/R)$ where $X_1, \dots, X_N$ are i.i.d. standard Gaussians and R is defined to be $(X_1^2+\dots+X_n^2)^{1/2}$ ? Or have you tried this already? Nov 20, 2009 at 8:19