The Lukacs theorem and the Olkin-Baker equation
Probability
2012-05-15 v2
Abstract
The Olkin-Baker functional equation is closely related to the celebrated Lukacs characterization of the gamma distribution. Its deeper understanding 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.
Keywords
Cite
@article{arxiv.1112.1226,
title = {The Lukacs theorem and the Olkin-Baker equation},
author = {Roman Ger and Jolanta Misiewicz and Jacek Wesolowski},
journal= {arXiv preprint arXiv:1112.1226},
year = {2012}
}