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=ξ12 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=ξ12 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 be a locally compact Abelian group. Denote by the convolution semigroup of probability distributions on , and by the degenerate distribution concentrated at . Let .

The Poisson distribution generated by the measure is defined as a shifted distribution of the form

One has the following

Raikov's theorem on locally compact Abelian groups

Let be the Poisson distribution generated by the measure . Suppose that , with . If is either an infinite order element, or has order 2, then is also a Poisson's distribution. In the case of being an element of finite order , can fail to be a Poisson's distribution.

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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.CS1 maint: multiple names: authors list (link)
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