View source for 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

:<math>P = P_1 \otimes P_2 </math>

where ''P''<sub>1</sub> is a probability distribution without any [[indecomposable distribution|indecomposable]] factor and ''P''<sub>2</sub> 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]]<ref>{{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}}</ref> for distributions on the line, and later it became clear<ref>{{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 }}</ref> that it is valid for distributions on considerably more general groups. A broad class (see<ref>D.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of -functions" ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'', '''9''' : 3 (1968) pp. 163–195</ref><ref>R. Davidson, "Arithmetic and other properties of certain Delphic semi-groups" ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'', '''10''' : 2 (1968) pp. 120–172</ref><ref>I.Z. Ruzsa, G.J. Székely, "Algebraic probability theory", Wiley (1988)</ref>) 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==
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[[Category:Theory of probability distributions]]


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