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] Contenu 1 Énoncé du théorème 2 Commentaire 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 (Linnik's theorem [ru]).

An extension to locally compact Abelian groups Let {style d'affichage X} be a locally compact Abelian group. Denote by {displaystyle M^{1}(X)} the convolution semigroup of probability distributions on {style d'affichage X} , and by {style d'affichage E_{X}} the degenerate distribution concentrated at {style d'affichage xin X} . Laisser {style d'affichage x_{0}en 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 {style d'affichage lui } be the Poisson distribution generated by the measure {displaystyle lambda E_{X_{0}}} . Supposer que {displaystyle mu =mu _{1}*dans _{2}} , avec {style d'affichage lui _{j}in M^{1}(X)} . Si {style d'affichage x_{0}} is either an infinite order element, or has order 2, alors {style d'affichage lui _{j}} is also a Poisson's distribution. In the case of {style d'affichage x_{0}} being an element of finite order {displaystyle nneq 2} , {style d'affichage lui _{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. L. (1970). "Certain statistical and probability problems on groups". Trudy Mat. Inst.. Steklov. 111: 52–109. ^ Linnik, Yu. V, Ostrovskii, je. V. (1977). Decomposition of random variables and vectors. Providence, R. JE.: Traductions de monographies mathématiques, 48. Société mathématique américaine. Catégories: Characterization of probability distributionsProbability theoremsTheorems in statistics