Random number distributions are characterized by various measures: probability moments, central and absolute moments, quantiles, mode, scattering, skewness, and excess (kurtosis) coefficients, and so on. All the ordinary sample characteristics converge in probability to the corresponding measures of distribution when the sample size tends to infinity [Cram46
]. Commonly, the characteristics based on the distribution moments are asymptotically normal with large sample sizes. Some classes of sample characteristics that are not based on sampling moments are also asymptotically normal, while others have quite different asymptotic behavior. When the limit probability distribution is known, you can build a statistical test to check whether a particular sample characteristic agrees with the corresponding measure of the distribution.