# Khinchin theorem

Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ admits (in the convolution semi-group of probability distributions) a factorization $$ P = P_1 \otimes P_2 \label{1} $$ 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.

#### References

[1] | A.Ya. Khinchin, "On the arithmetic of distribution laws" Byull. Moskov. Gos. Univ. Sekt. A , 1 : 1 (1937) pp. 6–17 (In Russian) |

[2] | K.R. Parthasarathy, R. Ranga Rao, S.R. Varadhan, "Probability distribution on locally compact Abelian groups" Illinois J. Math. , 7 (1963) pp. 337–369 |

[3] | D.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of $p$-functions" Z. Wahrscheinlichkeitstheor. Verw. Geb. , 9 : 3 (1968) pp. 163–195 |

[4] | R. Davidson, "Arithmetic and other properties of certain Delphic semi-groups" Z. Wahrscheinlichkeitstheor. Verw. Geb. , 10 : 2 (1968) pp. 120–172 |

[5] | I.Z. Ruzsa, G.J. Székely, "Algebraic probability theory" , Wiley (1988) |

#### Comments

A distribution of class $I_0$ is a distribution without indecomposable factor.

#### References

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

For Khinchin's theorem on Diophantine approximation see Diophantine approximation, metric theory of.

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Khinchin theorem.

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