# 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 $x_{0}∈R_{1}$ and $Φ_{a,σ}$ stands for the normal distribution $N(a,σ_{2})$; 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