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

### H.-J. Rossberg

Universität Leipzig, Germany

## Abstract

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

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.

## Cite this article

H.-J. Rossberg, Generalization of Cramér’s and Linnik’s Factorization Theorems in the Continuation Theory of Distribution Functions. Z. Anal. Anwend. 4 (1985), no. 3, pp. 193–200

DOI 10.4171/ZAA/144