Relations between the first four moments

TL;DR

This paper investigates inequalities relating the first four moments of a random variable, establishing bounds such as m_3  c m_4^{3/4} for variables with non-positive mean, advancing understanding of moment relationships.

Contribution

It provides new optimal bounds connecting the third and fourth moments for random variables with non-positive mean.

Findings

Established the inequality m_3  c m_4^{3/4} with the best possible constant c.

Identified conditions under which the inequality holds with equality.

Enhanced theoretical understanding of moment relationships in probability theory.

Abstract

Let m_j denote the jth moment of a random variable X. One of the results is the inequality m_3 \leq c m_4^{3/4} with the best possible factor c over all random variables X with a non-positive mean.