Khinchin's theorem on the factorization of distributions

Khinchin's theorem on the factorization of distributions says that every probability distribution P admits (in the convolution semi-group of probability distributions) a factorization

${\displaystyle P=P_{1}\otimes P_{2}}$

where P₁ is a probability distribution without any indecomposable factor and P₂ is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions. The factorization is not unique, in general.

The theorem was proved by A. Ya. Khinchin^[1] for distributions on the line, and later it became clear^[2] that it is valid for distributions on considerably more general groups. A broad class (see^[3][4][5]) of topological semi-groups is known, including the convolution semi-group of distributions on the line, in which factorization theorems analogous to Khinchin's theorem are valid.