Let's say I have a set of i.i.d. samples $X_1,\ldots,X_N \sim N_p(\mu, \Sigma)$. Now define \begin{equation} d^2_i(b)=(X_i - b)'\Sigma^{-1}(X_i-b) \end{equation} which is essentially the Mahalanobis distance, except that it measures the distance from $X_i$ to $b$ (with respect to the principal component axes), rather than the distance from $X_i$ to $\mu$, if that makes any sense.
I know that if $b=\mu$ then we would have $d^2_i\sim \chi^2_p$, but I'm having trouble figuring out what it would be otherwise. Any help would be greatly appreciated. Thanks!
