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

TL;DR

This paper provides an explicit bound on how closely a mixture of normal distributions approximates a normal distribution, with applications to phenotypic trait evolution in phylogenetics, demonstrating convergence as sample size increases.

Contribution

It introduces a new explicit bound on the Kolmogorov distance for mixtures of normals and applies it to evolutionary models, extending previous limit theorems.

Findings

Bound depends on first two moments of conditional moments

Distribution of average phenotypic trait converges to normal as n increases

Extends earlier limit theorems in phylogenetic models

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.