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arXiv · 2603.00073

A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities

Taehun Kim·Jung Chan Lee·ByoungSeon Choi
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Abstract

The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available until Kwon (2019, 2022) proposed analytic solutions for the valid region of Gram-Charlier densities. Despite the significance of the analytical solutions, the expressions remain algebraically complex. As these conditions for the Gram-Charlier densities are determined by a quartic polynomial, it is essential to investigate its positivity. In this work, necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed.

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Taehun Kim, Jung Chan Lee, ByoungSeon Choi. 2026-02-12. A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities. https://arxiv.org/abs/2603.00073

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