The Lukacs theorem and the Olkin-Baker equation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:399

Article

Ger, Roman²  Misiewicz, Jolanta¹  Wesolowski, Jacek¹ 

[1] Warsaw Univ Sci & Technol, Wydzial Matemat & Nauk Informacyjnych, Warsaw, Poland

[2] Uniwesytet Slqski, Inst Matemat, Katowice, Poland

关键词: Characterizations of probability distributions;    Gamma distribution;    Functional equations;    Additive function;    Logarithmic type function;    Semi-constant function;    Proper linearly invariant ideal;   

DOI  :  10.1016/j.jmaa.2012.10.034

来源: Elsevier

【 摘 要 】

The Olkin-Baker functional equation, except of being studied inside the theory of functional equations, is closely related to the celebrated Lukacs characterization of the gamma distribution. Its deeper understanding in the case of measurable unknown functions is essential to settle a challenging question of multivariate extensions of the Lukacs theorem. In this paper, first, we provide a new approach to the additive Olkin-Baker equation which holds almost everywhere on (0, infinity)(2) (with respect to the Lebesgue measure on R-2) under measurability assumption. Second, this new approach is adapted to the case when unknown functions are allowed to be non-measurable and the complete solution is given in such a general case. Third, the Olkin-Baker equation holding outside of a set from proper linearly invariant ideal of subsets of R-2 is considered. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free