The Lukacs theorem and the Olkin-Baker equation

Roman Ger; Jolanta Misiewicz; Jacek Wesolowski

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arXiv · 1112.1226

The Lukacs theorem and the Olkin-Baker equation

Roman Ger·Jolanta Misiewicz·Jacek Wesolowski

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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.

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Roman Ger, Jolanta Misiewicz, Jacek Wesolowski. 2012-05-13. The Lukacs theorem and the Olkin-Baker equation. https://arxiv.org/abs/1112.1226

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