View source for Indecomposable distribution

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A non-degenerate probability distribution that cannot be represented as a convolution of non-degenerate distributions. A random variable with an indecomposable distribution cannot be represented as a sum of independent non-constant random variables.

Examples of indecomposable distributions are the [[Arcsine distribution|arcsine distribution]], the [[Beta-distribution|beta-distribution]] when $n+m<2$, the [[Wishart distribution|Wishart distribution]], and any distribution in $\mathbf R^k$, $k\geq2$, that is concentrated on a strictly-convex closed hypersurface. The set of indecomposable distributions is sufficiently rich and is dense in the set of all distributions with the topology of weak convergence.

In the convolution semi-group of probability distributions the indecomposable ones play a role that is analogous, to a certain extent, to that of prime numbers in arithmetic (see [[Khinchin theorem|Khinchin theorem]] on the factorization of distributions), but not every distribution has indecomposable factors.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,   I.V. Ostrovskii,   "Decomposition of random variables and vectors" , Amer. Math. Soc.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.V. Ostrovskii,   "The arithmetic of probability distributions"  ''Theor. Probab. Appl.'' , '''31''' :  1  (1987)  pp. 1–24  ''Teor. Veroyatn. Primenen.'' , '''31'''  (1986)  pp. 3–30</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.R. Parthasarathy,   R.R. Rao,   S.R.S. Varadhan,   "On the category of indecomposable distributions on topological groups"  ''Trans. Amer. Soc.'' , '''102'''  (1962)  pp. 200–217</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,   "Characteristic functions" , Griffin  (1970)</TD></TR></table>