Assume we have a set of $N$ random variables with known multivariate distribution $\boldsymbol{X}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$, and a series of realisations $\{\boldsymbol{X_t}\}=\{x_{1,t},x_{2,t},\dots,x_{N,t}\}$ for $t=1,2,\dots,T$.
At each time time $t$ we can calculate the cross-sectional mean $\mu^{cs}_t=\frac{1}{N}\sum_{i=1}^N(x_{i,t})$ and the cross-sectional volatility: \begin{equation} \sigma^{cs}_t = \sqrt{\frac{1}{N}\sum_{i=1}^N(x_{i,t}-\mu^{cs}_t)^2}, \end{equation}
and hence we have the time series of cross-sectional values $\{\boldsymbol{\mu^{cs}_t}\}$ and $\{\boldsymbol{\sigma^{cs}_t}\}$.
Given the assumption of multivariate normality, is there an analytical solution for the limiting distribution of cross-sectional volatility (and mean)?
PS: the context is daily/weekly/monthly financial stock returns and then trying to model the cross-sectional volatility (a.k.a. return dispersion) over those same periods.
