The Lukacs theorem and the Olkin-Baker equation

TL;DR

This paper advances the understanding of the Olkin-Baker functional equation related to the Lukacs theorem, providing new solutions under various measurability conditions to aid in multivariate gamma distribution characterization.

Contribution

It introduces a novel approach to solving the Olkin-Baker equation almost everywhere, extends solutions to non-measurable functions, and considers solutions outside specific null sets.

Findings

New solutions for the Olkin-Baker equation under measurability assumptions

Complete solutions for non-measurable functions

Analysis of solutions outside linearly invariant null sets

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.