JournalszaaVol. 4, No. 3pp. 193–200

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

  • H.-J. Rossberg

    Universität Leipzig, Germany
Generalization of Cramér’s and Linnik’s Factorization Theorems in the Continuation Theory of Distribution Functions cover

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 F1F_1 and F2F_2 be two non-degenerate distribution functions such that

F1F2(x)=Φa,σ(x),xx0F_1 \ast F_2(x) = \Phi_{a, \sigma}(x), \quad \quad x \leq x_0

where x0R1x_0 \in \mathbb R_1 and Φa,σ\Phi_{a, \sigma} stands for the normal distribution N(a,σ2)N(a, \sigma^2); if the corresponding characteristic functions f1f_1 and f2f_2 do not vanish in the upper half plane, then F1F_1 and F2F_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