Raikov's theorem

Raikov's theorem 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.[1][2][3] Conteúdo 1 Declaração do teorema 2 Comente 3 An extension to locally compact Abelian groups 4 Raikov's theorem on locally compact Abelian groups 5 References 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 (teoria de Linnik [ru]).

An extension to locally compact Abelian groups Let {estilo de exibição X} be a locally compact Abelian group. Denote by {displaystyle M^{1}(X)} the convolution semigroup of probability distributions on {estilo de exibição X} , and by {estilo de exibição E_{x}} the degenerate distribution concentrated at {estilo de exibição xin X} . Deixar {estilo de exibição x_{0}in X,lambda >0} .

The Poisson distribution generated by the measure {displaystyle lambda E_{x_{0}}} is defined as a shifted distribution of the form {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 {mostre o estilo dele } be the Poisson distribution generated by the measure {displaystyle lambda E_{x_{0}}} . Suponha que {displaystyle mu =mu _{1}*dentro _{2}} , com {mostre o estilo dele _{j}in M^{1}(X)} . Se {estilo de exibição x_{0}} is either an infinite order element, or has order 2, então {mostre o estilo dele _{j}} is also a Poisson's distribution. In the case of {estilo de exibição x_{0}} being an element of finite order {displaystyle nneq 2} , {mostre o estilo dele _{j}} can fail to be a Poisson's distribution.

References ^ D. Raikov (1937). "On the decomposition of Poisson laws". Dokl. Acad. Sci. URSS. 14: 9-11. ^ Rukhin A. eu. (1970). "Certain statistical and probability problems on groups". Trudy Mat. Inst. Steklov. 111: 52-109. ^ Linnik, Yu. V., Ostrovskii, EU. V. (1977). Decomposition of random variables and vectors. Providência, R. EU.: Traduções de Monografias Matemáticas, 48. Sociedade Americana de Matemática. Categorias: Characterization of probability distributionsProbability theoremsTheorems in statistics