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

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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 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.

Related

Diophantine approximation