“Raikov's theorem”的意思、由来-开放百科全书

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 ({{ill|Linnik's theorem on convolution of normal distribution and Poisson's distribution|lt=Linnik's theorem|ru|Теорема Линника о разложении свертки нормального распределения и распределения Пуассона}}).

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

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.

References

1. ^{{cite journal|author=D. Raikov|first=|date=|year=1937|title=On the decomposition of Poisson laws|url=|journal=Dokl. Acad. Sci. URSS|volume=14|pages=9-11|via=}}
2. ^{{Cite journal|last=Rukhin A. L.|first=|date=1970|title=Certain statistical and probability problems on groups|url=|journal=Trudy Mat. Inst. Steklov|volume=111|pages=52-109|via=}}
3. ^{{Cite book|title=Decomposition of random variables and vectors|last=Linnik, Yu. V., Ostrovskii, I. V.|first=|publisher=Translations of Mathematical Monographs, 48. American Mathematical Society|year=1977|isbn=|location=Providence, R. I.|pages=}}