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

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.