Solve the following:

Denote S"-1 := {x E R" : [|x]| =1} the unit sphere in R". There exists a unique measure Un-1 on S"-, which is invariant with respect to the orthogonal transformations of R". That is, for every A c S"- Borel and orthogonal matrix H Un-1 (HA) = Un-1(A). This measure is called the Haar-measure on S"- which is essentially the uniform distribution on Sn-1 (a) Let X = (X1, . .., Xn) E R" be a vector valued random variable in R" such that X1, . .., Xn are i.i.d. random variables with distribution N(0, 1). Show that for any or- thogonal matrix H, the random vector Y := HX has i.i.d. components with distribution N(0, 1). Using the unicity of the Haar measure show that the distribution of X/| |X| is Un-1. That is, P(X/||XII E A) = un-1(A) for any Borel set A. (b) Let X1, X2, ... be i.i.d. with distribution N(0, 1), and let Rn = (X? + ... + X2) 1/2. Show that Rn/ Vn - 1 almost surely. (c) Let (Y1", .. (n), . . .. Ym") ) be a random vector in S"- with distribution vn-1. Using (a) and (b) show 1 lim P(Vny (")