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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.

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  • $\begingroup$ Are the $N$ random variables independent? $\endgroup$
    – Abdoul Haki
    Commented Aug 30, 2021 at 13:40
  • $\begingroup$ Maybe but not necessarily. I have changed the notation to showcase multivariate normal distribution with a general covariance matrix $\Sigma$ $\endgroup$
    – Emlyn Flint
    Commented Aug 30, 2021 at 14:16
  • $\begingroup$ @AbdoulHaki If Xs are stock returns then they can be safely assumed to be uncorrelated (though not independent) and their signs can be assumed to be independent in time as markets are quite efficient. $\endgroup$
    – Cagdas Ozgenc
    Commented Aug 30, 2021 at 15:49
  • $\begingroup$ @CagdasOzgenc that is not a safe assumption at all. Stock are returns are definitely correlated through time. $\endgroup$
    – Emlyn Flint
    Commented Aug 31, 2021 at 7:02
  • $\begingroup$ Stocks have contemporaneous correlation and almost nonexistent serial correlation. Please read carefully before you object. $\endgroup$
    – Cagdas Ozgenc
    Commented Aug 31, 2021 at 19:42

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