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Raikov's theorem

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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.

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 [ru]).

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 shifted distribution of the form

μ = e ( λ E x 0 ) = e − λ ( E 0 + λ E x 0 + λ 2 E 2 x 0 / 2 ! + … + λ n E n x 0 / n ! + … ) . {\displaystyle \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 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)} . If x 0 {\displaystyle x_{0}} is either an infinite order element, or has order 2, then μ j {\displaystyle \mu _{j}} is also a Poisson's distribution. In the case of x 0 {\displaystyle x_{0}} being an element of finite order n ≠ 2 {\displaystyle n\neq 2} , μ j {\displaystyle \mu _{j}} can fail to be a Poisson's distribution.

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