Über die Faktorisierung von Kompositionen einer abzählbaren Anzahl Poissonscher Gesetze.(Russian)Zbl 0358.60034

Consider an infinitely divisible distribution $$F$$ with characteristic function of the form (1) $$\varphi(t,F)=\exp\left\{i\beta t+\sum\limits_{\lambda}h(\lambda)(e^{i\lambda t}-1)\right\}$$ where $$h$$ is a non-negative function and $$\sum\limits_{\lambda}h(\lambda)<+\infty$$. The author gives necessary and sufficient conditions for $$F$$ to belong to the class $$I_0$$ of those infinitely divisible distributions for which all factors are also infinitely divisible. Let $$D(h)= \{\lambda:h(\lambda)\neq0\}$$. By $$M(A)(M^+(A))$$ we denote the set of all finite linear combinations with integer (natural) coefficients of elements of $$A$$. The author proves: Theorem 1. Let $$F$$ be an infinitely divisible distribution with characteristic function of the form (1) satisfying the conditions a) $$D(h)\subset Q^+$$, where $$Q^+$$ is the set of positive rational numbers, b) for some $$K>0\sum\limits_{|\lambda|>y}h(\lambda)=0(\exp(-Ky^{2})),y\to\infty$$. Then $$F$$ belongs to $$I_0$$ if and only if for all $$p\in M^+(D(h))+M^+(D(h))$$ the following is true: $$p\notin M(D(h)\cap(p,\infty))$$. This result is a generalization of a theorem of P. Lévy where it has been supposed that $$D(h)$$ is a finite subset of the set of natural numbers.
Reviewer: E. Warmuth

MSC:

 6e+06 Probability distributions: general theory