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=ξ12 has a Poisson distribution as well. It turns out that the converse is also valid.[1][2][3]

Statement of the theorem

edit

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

edit

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

edit

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

edit

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.

References

edit
  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)