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, the beta-distribution when $n+m<2$, the 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 on the factorization of distributions), but not every distribution has indecomposable factors.
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Indecomposable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indecomposable_distribution&oldid=33508