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.

翻译：Gram-Charlier概率密度函数的正定性数十年来一直是广泛研究的课题。自Barton和Dennis (1952)引入数值正定性条件以来，直到Kwon (2019, 2022)才提出Gram-Charlier密度有效区域的解析解。尽管解析解意义重大，但其表达式仍具代数复杂性。由于Gram-Charlier密度的条件由四次多项式决定，因此研究其正定性至关重要。本文通过分离方法推导了四次多项式正定性的充分必要条件，并基于这些条件提出了Gram-Charlier密度正定性的更简洁解析表达式。