Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

Abstract: An explicit bound is given for the Kolmogorov distance between a mixture of normal distributions and a normal distribution with properly chosen parameter values. A random variable X has a mixture of normal distributions if its conditional distribution given some sigma-algebra is normal. The bound depends only on the first two moments of the first two conditional moments of X given this sigma-algebra. As an application, the Yule-Ornstein-Uhlenbeck model, used in the field of phylogenetic comparative methods, is considered. A bound is derived for the Kolmogorov distance between the distribution of the average value of a phenotypic trait over n related species and a normal distribution. The bound goes to 0 as n goes to infinity, extending earlier limit theorems by Bartoszek and Sagitov.