Let $(X_1, \cdots, X_n)$ be a random vector uniformly distributed on the sphere \(X_1^2 + \cdots + X_n^2 = n.\) Find the limiting distribution of $(X_1, X_2, X_3)$ as $n \rightarrow \infty$.

Solution:

Let $S_n(R)$ denote the area of the $n$-dimensional sphere or radius $R$. We know that

\[S_n(R) = \frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2} \right)} R^{n}.\]

Hence, the density of $(X_1, \cdots, X_n)$ is given by:

\[f(x_1,\cdots,x_n) = \begin{cases} \frac{1}{S_n(\sqrt{n})}, &\text{ if } x_1^2 + \cdots + x_n^2 = n\\ 0, &\text{ otherwise } \end{cases}.\]

Therefore, the joint density of $(X_1, X_2, X_3)$ can be computed as follows:

\[\begin{aligned} f_{X_1,X_2,X_3} (x_1,\cdots,x_3) & = \idotsint\limits_{\mathbb{R}^{n-3}} f(x_1,\cdots,x_n) ~dx_4\cdots dx_n \\ & = \idotsint\limits_{x_1^2 + \cdots + x_n^2 = n} \frac{1}{S_n(\sqrt{n})}~dx_4\cdots dx_n \\ & = \frac{1}{S_n(\sqrt{n})} \idotsint\limits_{x_4^2 + \cdots + x_n^2 = n-x_1^2-x_2^2-x_3^2}~dx_4\cdots dx_n \\ & = \frac{S_{n-3}(\sqrt{n-x_1^2-x_2^2-x_3^2})}{S_n(\sqrt{n})}\\ & = \frac{\frac{2\pi^{\frac{n-2}{2}}}{\Gamma\left(\frac{n-2}{2} \right)} (n-x_1^2-x_2^2-x_3^2)^{\frac{n-3}{2}}}{\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2} \right)} n^{\frac{n}{2}}}\\ & = \frac{\pi^{-\frac{3}{2}}\Gamma\left(\frac{n+1}{2} \right)}{(n-x_1^2-x_2^2-x_3^2)^{\frac{3}{2}}\Gamma\left(\frac{n-2}{2} \right)} \left(1- \frac{x_1^2+x_2^2+x_3^2}{n}\right)^{\frac{n}{2}}. \end{aligned}\]

Since

\[\frac{\Gamma(s+\alpha)}{\Gamma(s) } \sim s^{\alpha}, \quad \alpha \in \mathbb{C},\]

as $\left\vert s \right\vert \rightarrow \infty$, the first term of the above quantity should goes to some constant $C$ that doesn’t depend on $x_1, x_2, x_3$.

For the second term, we see that

\[\lim_{n\rightarrow \infty} \left(1- \frac{x_1^2+x_2^2+x_3^2}{n}\right)^{\frac{n}{2}} = \lim_{n\rightarrow \infty} \left(1- \frac{x_1^2+x_2^2+x_3^2}{\left(\frac{n}{2}\right) 2}\right)^{\frac{n}{2}} = e^{-\frac{x_1^2+x_2^2+x_3^2}{2}}.\]

Therefore, we get

\[\begin{aligned} \lim_{n\rightarrow \infty} f_{X_1,X_2,X_3} (x_1,\cdots,x_3) = C e^{-\frac{x_1^2+x_2^2+x_3^2}{2}}, \end{aligned}\]

which is the density of three i.i.d. standard normal random variable.

Remarks:

  • It’s easy to verify that $C= (2\pi)^{-3/2}$ so the limiting distribution exists.
  • The statement is also true if $(X_1, \cdots, X_n)$ is uniformly distributed inside the $n$-dim. ball of radius $\sqrt{n}$.
  • This result is a simpler version of Poincaré’s Lemma. If we fix $k$ and look at the first $k$ coordinates, then we will get $k$ i.i.d standard normal r.v.’s.