User:Serg.sinel/sandbox

Linnik's theorem on decomposing a convolution of normal and Poisson's distributions is a result in probability theory. By a virtue of Cramér's decomposition theorem, if a sum of two independent random variables is normally distributed, then both summands are normally distributed as well. A similar claim is also valid for Poisson's distribution (Raikov's theorem). Linnik's theorem establishes a similar property for convolution of normal and Poisson's distributions.

Statement of the theorem
Suppose ξ is a random variable whose distribution is a convolution of normal and Poisson's distributions, and that ξ is a sum of two independent random variables, ξ=ξ1+ξ2. Then the distribution of each summand (ξ1 and ξ2) is also a convolution of normal and Poisson's distrubutions.

Note
The Linnik's theorem implies that a convolution of normal and Poisson's distributions is in the Linnik class I0, that is, such convolution admits no indecomposable divisors.