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

where P1 is a probablity 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[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][4][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. ↑ Kinchin, A. Ya. (1937). On the arithmetic of distribution laws (in Russian). Byull. Moskov. Gos. Univ. Sekt. pp. 6–17.
  2. ↑ Parthasarathy, K.R.; Ranga Rao, R.; Varadhan, S.R. (1963). Probability distribution on locally compact Abelian groups. Illinois J. Math. pp. 337–369.
  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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