Characterization of probability distributions

Characterization theorems in probability theory and mathematical statistics are such theorems that establish a connection between the type of the distribution of random variables or random vectors and certain general properties of functions in them. For example, according to G. Polya's characterization theorem, if $X_{1}$ and $X_{2}$ are independent identically distributed random variables with finite variance, then statistics $S_{1}=X_{1}$ and $S_{2}={\frac {(X_{1}+X_{2})}{\sqrt {2}}}$ are identically distributed if and only if $X_{1}$ and $X_{2}$ have the normal distribution with zero mean. The assumption that two linear (or non-linear) statistics are identically distributed (or independent, or have a constancy regression and so on) can be used to characterize various populations (see, for example, ).