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] Inhalt 1 Aussage des Theorems 2 Kommentar 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 {Anzeigestil X} be a locally compact Abelian group. Bezeichne mit {displaystyle M^{1}(X)} the convolution semigroup of probability distributions on {Anzeigestil X} , and by {Anzeigestil E_{x}} the degenerate distribution concentrated at {Anzeigestil xin X} . Lassen {Anzeigestil 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!+Punkte ).} One has the following Raikov's theorem on locally compact Abelian groups Let {zeige ihn an } be the Poisson distribution generated by the measure {displaystyle lambda E_{x_{0}}} . Nehme an, dass {displaystyle mu =mu _{1}*in _{2}} , mit {displaystyle ihn _{j}in M^{1}(X)} . Wenn {Anzeigestil x_{0}} is either an infinite order element, or has order 2, dann {displaystyle ihn _{j}} is also a Poisson's distribution. In the case of {Anzeigestil x_{0}} being an element of finite order {displaystyle nneq 2} , {displaystyle ihn _{j}} can fail to be a Poisson's distribution.

References ^ D. Raikov (1937). "On the decomposition of Poisson laws". Dokl. Akad. Wissenschaft. 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, ich. v. (1977). Decomposition of random variables and vectors. Vorsehung, R. ICH.: Übersetzungen mathematischer Monographien, 48. Amerikanische Mathematische Gesellschaft. Kategorien: Characterization of probability distributionsProbability theoremsTheorems in statistics