# Sample distribution of cross-sectional statistics of returns

Currently doing an application of VaR on sample of industry portfolios in the US. I have a matrix of $$n$$ industry portfolios with $$m$$ time-series observations. I calculate cross-sectionally (for each trading day) the mean,median, std.dev and kurtosis of the sample $$(r_{m,1},r_{m,2},...,r_{m,n})$$. For each cross-section sample, we have one estimator. For the whole time-period we have $$m$$ observation of the cross-sectional statistics.

My question is:

Is the distribution of these cross-sectional statistics (let $$\hat{\theta_t}$$ denote the estimator for the sample at time t) a sampling distribution?

If this holds, what information could be extracted from the sample distribution (e.g "the mean of the means")