Raikov's theorem

Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in the probability theory. It is well known that if each of two independent random variables $\xi _{1}$ and $\xi _{2}$ has a Poisson distribution, then their sum $\xi =\xi _{1}+\xi _{2}$ has a Poisson distribution as well. It turns out that the converse is also valid.

Statement of the theorem

Suppose that a random variable $\xi$ has a Poisson distribution and admits a decomposition as a sum $\xi =\xi _{1}+\xi _{2}$ of two independent random variables. Then the distribution of each summand is a shifted Poisson distribution.

Raikov's theorem is similar to Cramér’s decomposition theorem. The latter result claims that if a sum of two independent random variables has a 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). It follows from the Raikov theorem that the Poisson distribution belongs to the Linnik class $I_{0}$.

An extension to locally compact Abelian groups

Let $X$ be a locally compact Abelian group. Denote by $M^{1}(X)$ the convolution semigroup of probability distributions on $X$, and by $E_{x}$the degenerate distribution concentrated at $x\in X$. Let $x_{0}\in X,\lambda >0$.

The Poisson distribution generated by the measure $\lambda E_{x_{0}}$ is defined as a distribution of the form

$\mu =e(\lambda E_{x_{0}})=e^{-\lambda }\left(E_{0}+\lambda E_{x_{0}}+{\frac {\lambda ^{2}E_{2x_{0}}}{2!}}+\cdots +{\frac {\lambda ^{n}E_{nx_{0}}}{n!}}+\cdots \right).$

Theorem Let $\mu$ be the Poisson distribution generated by the measure $\lambda E_{x_{0}}$. Suppose that $\mu =\mu _{1}*\mu _{2}$, with $\mu _{j}\in M^{1}(X)$. Then each of $\mu _{j}$ is a shift of a Poisson distribution if and only if $x_{0}$ is either an infinite-order element or has order 2.