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

:$P\; =\; P\_1\; \backslash otimes\; P\_2$

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{{Cite book|title=On the arithmetic of distribution laws|last=Kinchin|first=A. Ya.|publisher=Byull. Moskov. Gos. Univ. Sekt.|year=1937|pages=6–17|language=ru}} for distributions on the line, and later it became clear{{cite journal |last1=Parthasarathy |first1=K. R. |author-link1=K. R. Parthasarathy (probabilist) |last2=Rao |first2=R. Ranga |author-link2=R. Ranga Rao |last3=Varadhan |first3=S. R. S. |author-link3=S. R. Srinivasa Varadhan |date=1963-06-01 |df=dmy-all |title=Probability distributions on locally compact Abelian groups |journal=Illinois Journal of Mathematics |volume=7 |issue=2 |pages=337–369 |doi=10.1215/ijm/1255644642 |doi-access=free }} that it is valid for distributions on considerably more general groups. A broad class (seeD.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of -functions" Z. Wahrscheinlichkeitstheor. Verw. Geb., 9 : 3 (1968) pp. 163–195R. Davidson, "Arithmetic and other properties of certain Delphic semi-groups" Z. Wahrscheinlichkeitstheor. Verw. Geb., 10 : 2 (1968) pp. 120–172I.Z. Ruzsa, G.J. Székely, "Algebraic probability theory", Wiley (1988)) 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.