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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")

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