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 One has the followingRaikov's theorem on locally compact Abelian groups
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.References
{{reflist Characterization of probability distributions Probability theorems Theorems in statistics