Raikov's theorem

Template:Short description 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 (Template:Ill). 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

$\mu =e(\lambda E_{x_0})=e^{-\lambda }(E_0+\lambda E_{x_0}+\frac{{\lambda }^2{E}_{2x_0}}{2!}+\cdots +\frac{{\lambda }^n{E}_{n{x}_0}}{n!}+\cdots ).$

Theorem Let $\mu$ be the Poisson distribution generated by the measure $\lambda E_{x_0}$. Suppose that $\mu ={\mu }_1\ast {\mu }_2$, 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.