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Khinchin's theorem on the factorization of distributions: Any probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553901.png" /> admits (in the convolution semi-group of probability distributions) a factorization
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Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ admits (in the convolution semi-group of probability distributions) a factorization
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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P = P_1 \otimes P_2 \label{1}
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553903.png" /> is a distribution of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553904.png" /> (see [[Infinitely-divisible distributions, factorization of|Infinitely-divisible distributions, factorization of]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553905.png" /> is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions (cf. [[Indecomposable distribution|Indecomposable distribution]]). The factorization (*) is not unique, in general.
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where $P_1$ is a distribution of class $I_0$ (see [[Infinitely-divisible distributions, factorization of]]) and $P_2$ is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions (cf. [[Indecomposable distribution]]). The factorization (1) is not unique, in general.
  
 
The theorem was proved by A.Ya. Khinchin [[#References|[1]]] for distributions on the line, and later it became clear [[#References|[2]]] that it is valid for distributions on considerably more general groups. A broad class (see [[#References|[3]]]–[[#References|[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.
 
The theorem was proved by A.Ya. Khinchin [[#References|[1]]] for distributions on the line, and later it became clear [[#References|[2]]] that it is valid for distributions on considerably more general groups. A broad class (see [[#References|[3]]]–[[#References|[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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "On the arithmetic of distribution laws"  ''Byull. Moskov. Gos. Univ. Sekt. A'' , '''1''' :  1  (1937)  pp. 6–17  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.R. Parthasarathy,  R. Ranga Rao,  S.R. Varadhan,  "Probability distribution on locally compact Abelian groups"  ''Illinois J. Math.'' , '''7'''  (1963)  pp. 337–369</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.G. Kendall,  "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553906.png" />-functions"  ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''9''' :  3  (1968)  pp. 163–195</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Davidson,  "Arithmetic and other properties of certain Delphic semi-groups"  ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''10''' :  2  (1968)  pp. 120–172</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.Z. Ruzsa,  G.J. Székely,  "Algebraic probability theory" , Wiley  (1988)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "On the arithmetic of distribution laws"  ''Byull. Moskov. Gos. Univ. Sekt. A'' , '''1''' :  1  (1937)  pp. 6–17  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  K.R. Parthasarathy,  R. Ranga Rao,  S.R. Varadhan,  "Probability distribution on locally compact Abelian groups"  ''Illinois J. Math.'' , '''7'''  (1963)  pp. 337–369</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  D.G. Kendall,  "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of $p$-functions"  ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''9''' :  3  (1968)  pp. 163–195</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  R. Davidson,  "Arithmetic and other properties of certain Delphic semi-groups"  ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''10''' :  2  (1968)  pp. 120–172</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  I.Z. Ruzsa,  G.J. Székely,  "Algebraic probability theory" , Wiley  (1988)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
A distribution of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055390/k0553907.png" /> is a distribution without indecomposable factor.
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A distribution of class $I_0$ is a distribution without indecomposable factor.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,  "Characteristic functions" , Griffin  (1970)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,  "Characteristic functions" , Griffin  (1970)</TD></TR>
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</table>
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For Khinchin's theorem on Diophantine approximation see [[Diophantine approximation, metric theory of]].
  
For Khinchin's theorem on Diophantine approximation see [[Diophantine approximation, metric theory of|Diophantine approximation, metric theory of]].
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Latest revision as of 16:26, 9 April 2016

Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ admits (in the convolution semi-group of probability distributions) a factorization $$ P = P_1 \otimes P_2 \label{1} $$ where $P_1$ is a distribution of class $I_0$ (see Infinitely-divisible distributions, factorization of) and $P_2$ is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions (cf. Indecomposable distribution). The factorization (1) 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][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] A.Ya. Khinchin, "On the arithmetic of distribution laws" Byull. Moskov. Gos. Univ. Sekt. A , 1 : 1 (1937) pp. 6–17 (In Russian)
[2] K.R. Parthasarathy, R. Ranga Rao, S.R. Varadhan, "Probability distribution on locally compact Abelian groups" Illinois J. Math. , 7 (1963) pp. 337–369
[3] D.G. Kendall, "Delphic semi-groups, infinitely divisible phenomena, and the arithmetic of $p$-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)


Comments

A distribution of class $I_0$ is a distribution without indecomposable factor.

References

[a1] E. Lukacs, "Characteristic functions" , Griffin (1970)

For Khinchin's theorem on Diophantine approximation see Diophantine approximation, metric theory of.

How to Cite This Entry:
Khinchin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Khinchin_theorem&oldid=38559
This article was adapted from an original article by I.V. Ostrovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article