If $X \sim \mathcal N(0,\Sigma)$, what is the distribution of $X$ conditioned on the event $\|X\| \le r$?

Question

Let $\Sigma$ be a $p$-by-$p$ positive-definite matrix and let $r > 0$. Let $X \sim \mathcal N(0,\Sigma)$ and $R = R(X):= \|X\|$.

Quesion. What is the distribution of $X$ conditioned on the event $R \le r$ ?

Answer 1

By Bayes' Theorem we can write:

$p(X | \| X \| \le r) \propto p(\| X \| \le r | X) p(X) = 1_{\{ \| X \| \le r\}} (x) \mathcal{N}(0, \Sigma)(x)$.

If $X$ is given and $r$ is a given constant, then there is nothing random in the expression $p(\| X \| \le r | X)$, so this is just an indicator function $1_{\{ \| X \| \le r\}}$ equal to 1 if $\| X \| \le r$ and 0 otherwise, so the distribution is one proportional to a normal distribution times the indicator function, making the distribution a truncated multivariate normal distribution.