# Limit of Hotelling $T^2$ distribution

Suppose you have a sample $X_1, \ldots, X_n$, $n$ large, from a multivariate normal distribution $N_p(\mu, \Sigma)$. It is easy to show that $D_i := k(X_i - \bar{X}_n)'S^{1}_n(X_i-\bar{X}_n) \sim T^2_{p, n-1}$, for some constant $k$ and $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ and $S_n = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X}_n)(X_i - \bar{X}_n)'$.

What is the distribution of $D_i$ when $n\rightarrow \infty$?

At a first sight I think $D_i \rightarrow_d \chi^2_{p}$ because $S^{-1} \rightarrow_p \Sigma$.