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 ξ 1 {\displaystyle \xi _{1}} and ξ 2 {\displaystyle \xi _{2}} has a Poisson distribution, then their sum ξ = ξ 1 + ξ 2 {\displaystyle \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 ξ {\displaystyle \xi } has a Poisson distribution and admits a decomposition as a sum ξ = ξ 1 + ξ 2 {\displaystyle \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 {\displaystyle I_{0}} .
An extension to locally compact Abelian groups Let X {\displaystyle X} be a locally compact Abelian group. Denote by M 1 ( X ) {\displaystyle M^{1}(X)} the convolution semigroup of probability distributions on X {\displaystyle X} , and by E x {\displaystyle E_{x}} the degenerate distribution concentrated at x ∈ X {\displaystyle x\in X} . Let x 0 ∈ X , λ > 0 {\displaystyle x_{0}\in X,\lambda >0} . The Poisson distribution generated by the measure λ E x 0 {\displaystyle \lambda E_{x_{0}}} is defined as a distribution of the form
Theorem Let μ {\displaystyle \mu } be the Poisson distribution generated by the measure λ E x 0 {\displaystyle \lambda E_{x_{0}}} . Suppose that μ = μ 1 ∗ μ 2 {\displaystyle \mu =\mu _{1}*\mu _{2}} , with μ j ∈ M 1 ( X ) {\displaystyle \mu _{j}\in M^{1}(X)} . Then each of μ j {\displaystyle \mu _{j}} is a shift of a Poisson distribution if and only if x 0 {\displaystyle x_{0}} is either an infinite-order element or has order 2.
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