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

Published online by Cambridge University Press:  17 March 2021

Krzysztof Bartoszek*
Affiliation:
Linköping University
Torkel Erhardsson*
Affiliation:
Linköping University
*
*Postal address: Department of Computer and Information Science, Linköping University, SE-581 83 Linköping, Sweden.
**Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden. Email address: [email protected]

Abstract

Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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