Raikov's theorem
Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in the mathematical theory of probability. It is well known that if each of two independent random variables ξ1 and ξ2 has a Poisson distribution, then their sum ξ = ξ1 + ξ2 has a Poisson distribution as well. It turns out that the converse is also valid.
Statement of the theorem
Suppose that a random variable ξ has Poisson's distribution and admits a decomposition as a sum ξ = ξ1 + ξ2 of two independent random variables. Then the distribution of each summand is a shifted Poisson's distribution.Comment
Raikov's theorem is similar to Cramér’s decomposition theorem. The latter result claims that if a sum of two independent random variables has normal distribution, then each summand is normally distributed as well. It was also proved by Yu. V. Linnik that a convolution of normal distribution and Poisson's distribution possesses a similar property.An extension to locally compact Abelian groups
Let be a locally compact Abelian group. Denote by the convolution semigroup of probability distributions on, and by the degenerate distribution concentrated at. Let.The Poisson distribution generated by the measure is defined as a shifted distribution of the form
Let be the Poisson distribution generated by the measure. Suppose that, with. If is either an infinite-order element, or has order 2, then is also a Poisson's distribution. In the case of being an element of finite order, can fail to be a Poisson's distribution.