Khinchin's theorem on the factorization of distributions

Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ admits (in the convolution semi-group of probability distributions) a factorization

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

where $P_{1}$ is a distribution of class $I_{0}$ (see Infinitely-divisible distributions, factorization of) and $P_{2}$ is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions (cf. Indecomposable distribution). The factorization (1) 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]–[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.

Comments

A distribution of class $I_{0}$ is a distribution without indecomposable factor.

References

[a1]	E. Lukacs, "Characteristic functions" , Griffin (1970)

Related

Diophantine_approximation#Khinchin.27s_theorem_and_extensions