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]\displaystyle{ P = P_1 \otimes P_2 }[/math]
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[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
- ↑ Kinchin, A. Ya. (1937) (in ru). On the arithmetic of distribution laws. Byull. Moskov. Gos. Univ. Sekt.. pp. 6–17.
- ↑ Parthasarathy, K. R.; Rao, R. Ranga; Varadhan, S. R. S. (1963-06-01). "Probability distributions on locally compact Abelian groups". Illinois Journal of Mathematics 7 (2): 337–369. doi:10.1215/ijm/1255644642.
- ↑ D.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of -functions" Z. Wahrscheinlichkeitstheor. Verw. Geb., 9 : 3 (1968) pp. 163–195
- ↑ R. Davidson, "Arithmetic and other properties of certain Delphic semi-groups" Z. Wahrscheinlichkeitstheor. Verw. Geb., 10 : 2 (1968) pp. 120–172
- ↑ I.Z. Ruzsa, G.J. Székely, "Algebraic probability theory", Wiley (1988)
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