Raikov's theorem

Raikov’s theorem 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 (Linnik's theorem).

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 shifted distribution of the form

$\mu =e(\lambda E_{x_{0}})=e^{-\lambda }(E_{0}+\lambda E_{x_{0}}+\lambda ^{2}E_{2x_{0}}/2!+\ldots +\lambda ^{n}E_{nx_{0}}/n!+\ldots ).$

One has the following

Raikov's theorem on locally compact Abelian groups

Let $\mu$ be the Poisson distribution generated by the measure $\lambda E_{x_{0}}$ . 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 $n\neq 2$ , $\mu _{j}$ can fail to be a Poisson's distribution.

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References

D. Raikov (1937). "On the decomposition of Poisson laws". Dokl. Acad. Sci. URSS. 14: 9–11.
Rukhin A. L. (1970). "Certain statistical and probability problems on groups". Trudy Mat. Inst. Steklov. 111: 52–109.
Linnik, Yu. V., Ostrovskii, I. V. (1977). Decomposition of random variables and vectors. Providence, R. I.: Translations of Mathematical Monographs, 48. American Mathematical Society.CS1 maint: multiple names: authors list (link)

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[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] Linnik, Yu. V., Ostrovskii, I. V. (1977). Decomposition of random variables and vectors. Providence, R. I.: Translations of Mathematical Monographs, 48. American Mathematical Society.CS1 maint: multiple names: authors list (link)