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Khinchin's theorem on the factorization of distributions

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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 ⊗ P 2 {\displaystyle P=P_{1}\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 for distributions on the line, and later it became clear that it is valid for distributions on considerably more general groups. A broad class (see) 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. Kinchin, A. Ya. (1937). On the arithmetic of distribution laws (in Russian). Byull. Moskov. Gos. Univ. Sekt. pp. 6–17.
  2. Parthasarathy, K. R.; Rao, R. Ranga; Varadhan, S. R. S. (1 June 1963). "Probability distributions on locally compact Abelian groups". Illinois Journal of Mathematics. 7 (2): 337–369. doi:10.1215/ijm/1255644642.
  3. D.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of -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)


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