The equations for the higher‐order moments of turbulent velocity fluctuations are considered. These are derived utilizing truncated, cumulant expansions as an approximation for the probability density distributions of the corresponding turbulence properties. By applying different degrees of truncations to these expansions, an alternative set of equations for the moments is formulated that contains only velocity correlations. From these equations, interrelations between the higher‐order moments are deduced and are experimentally verified using data available in the literature and also data measured by the authors.