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.
[1][2][3] 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.
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 [ru]).
be a locally compact Abelian group.
the convolution semigroup of probability distributions on
the degenerate distribution concentrated at
, λ > 0
The Poisson distribution generated by the measure
λ
is defined as a shifted distribution of the form
μ = e ( λ
be the Poisson distribution generated by the measure
Suppose that
is either an infinite order element, or has order 2, then
is also a Poisson's distribution.
being an element of finite order
can fail to be a Poisson's distribution.