Law of Large Numbers for whole distributions I'm aware of the Law(s) of Large Numbers, concerning the means. However, intuitively, I'd expect not just the mean, but also the observed relative frequencies (or the histogram, if we have a continuous distribution) to approach the theoretical PMF/PDF as the number of trials goes into infinity.

*

*Is my intuition wrong? Always or only for some degenerate cases (e.g. Cauchy)?

*If not, is there a special name for that law?


You may be asking for the Glivenko-Cantelli theorem.
https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem
Note that this is about cumulative distribution functions, i.e., for data sets relative frequencies that observations are below any given value $x$, or by implication in any interval $[x_1,x_2]$. Histograms are not a standard basis for such a result, as there are many ways to set up a histogram (bar widths etc.), whereas the cdf is uniquely defined.
