# 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?

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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? –  Yemon Choi Nov 20 '09 at 8:19

There has been some work in the physics community on extreme statistics (i.e. distribution of largest and smallest components) of random vectors. See, link text for example. The largest component is approximately distributed like a Gumbel random variable, while the smallest component is approximately distributed like an exponential random variable.

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