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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:

60E05 Probability distributions: general theory
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