Indecomposable distribution
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 , the Wishart distribution, and any distribution in
,
, 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.
References
[1] | Yu.V. Linnik, I.V. Ostrovskii, "Decomposition of random variables and vectors" , Amer. Math. Soc. (1977) (Translated from Russian) |
[2] | 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 |
[3] | 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 |
Comments
References
[a1] | E. Lukacs, "Characteristic functions" , Griffin (1970) |
Indecomposable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indecomposable_distribution&oldid=33508