Raikov's theorem

Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in probability theory. 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 (Linnik's theorem).

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 following

Raikov'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.