=+29.9. 20.261 A theorem of Poincar. (a) Suppose that X, = (X ,.,..., X ... ) is

=+29.9. 20.261 A theorem of Poincaré.

(a) Suppose that X, = (X ,.,..., X ... ) is uniformly distributed over the surface of a sphere of radius in R". Fix t, and show that X ,.,..., X ,,, are in the limit independent, each with the standard normal distribution. Hint: If the components of Y ,, - (Y ,., ..., Y ... )

are independent, each with the standard normal distribution, then X ,, has the same distribution as Vn Y ,, /|Y ,, I.