Raikov's theorem explained

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]}

Statement of the theorem

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.

Comment

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 .

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, λ > 0 .

The Poisson distribution generated by the measure

λ E _{x _{0}} is defined as a shifted distribution of the form\mu = e( λ E _{x _{0}} ) = e ^{-λ} (E _{0+λ} E _{x _{0}} +λ ^{2} E _{2 x _{0}} / 2 ! + \ldots +λ ^{n} E _{nx _{0}} /n! + \ldots).

One has the following

Raikov's theorem on locally compact Abelian groups

Let

\mu be the Poisson distribution generated by the measure λ E _{x _{0}} . Suppose that \mu = \mu _{1*\mu} _{2} , with \mu _{j\in} M ^{1(X)} . If x _{0} is either an infinite order element, or has order 2, then \mu _{j} is also a Poisson's distribution. In the case of x _{0} being an element of finite order n\ne 2 , \mu _{j} can fail to be a Poisson's distribution.

Notes and References

1. D. Raikov. 1937. On the decomposition of Poisson laws. Dokl. Acad. Sci. URSS. 14. 9–11.
2. Rukhin A. L.. 1970. Certain statistical and probability problems on groups. Trudy Mat. Inst. Steklov. 111. 52–109.
3. Book: Linnik, Yu. V., Ostrovskii, I. V.. Decomposition of random variables and vectors. Translations of Mathematical Monographs, 48. American Mathematical Society. 1977. Providence, R. I..