This is one of exercises in Probability: theory and examples, Durrett
3.2.15.
Show that if $X_n = (X^1_n, …,X^n_n) $ is uniformly distributed over
the surface of the sphere of radius $\sqrt{n}$ in $R^n$, then $X^1_n \to $ a standard normal in distribution.Hint: Let $Y_1, Y_2, …$ be i.i.d. standard normals an let $X^i_n = Y_i (n/ \sum^n_{m=1} Y^2_m)^{1/2} $.
Here's my question:
How to prove this r.v $~Y_i (n/ \sum^n_{m=1} Y^2_m)^{1/2}~$ is uniformly distributed over the surface of the sphere of radius $\sqrt{n}$?
Or are there any other approaches to solve this problem?
Best Answer
One is supposed to use the fact that the distribution of the vector $Y=(Y_m)_{1\leqslant m\leqslant n}$ is invariant by the rotations, since its density depends only on $R=\left(\sum\limits_{m=1}^nY_m^2\right)^{1/2}$. These rotations leave $R$ invariant hence the distribution of $U=Y\cdot\sqrt{n/R}$ is invariant by the rotations and $U$ is indeed uniformly distributed on the sphere of radius $\sqrt{n}$.
Now, the strong law of large numbers indicates that $R^2/n\to E(Y_1^2)=1$ almost surely. Obviously, $Y_1$ converges in distribution to a standard normal distribution hence $U_1=Y_1\cdot\sqrt{n/R}$ converges in distribution to a standard normal distribution.