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Zeitschrift für Analysis und ihre Anwendungen


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Volume 4, Issue 3, 1985, pp. 193–200
DOI: 10.4171/ZAA/144

Published online: 1985-06-30

Generalization of Cramér’s and Linnik’s Factorization Theorems in the Continuation Theory of Distribution Functions

H.-J. Rossberg[1]

(1) Universität Leipzig, Germany

Stimulated by a problem of Kruglov and a result of Titov we derive an elementary continuation theorem for distribution functions. It implies the following generalization of Carmér’s theorem. Let $F_1$ and $F_2$ be two non-degenerate distribution functions such that $$F_1 \ast F_2(x) = \Phi_{a, \sigma}(x), \quad \quad x \leq x_0$$ where $x_0 \in \mathbb R_1$ and $\Phi_{a, \sigma}$ stands for the normal distribution $N(a, \sigma^2)$; if the corresponding characteristic functions $f_1$ and $f_2$ do not vanish in the upper half plane, then $F_1$ and $F_2$ are also normal. Linnik’s theorem can be analogously generalized. More general variants are also discussed.

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Rossberg H.-J.: Generalization of Cramér’s and Linnik’s Factorization Theorems in the Continuation Theory of Distribution Functions. Z. Anal. Anwend. 4 (1985), 193-200. doi: 10.4171/ZAA/144