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

Part of: Limit theorems Distribution theory

Published online by Cambridge University Press:  17 March 2021

Krzysztof Bartoszek  and
Torkel Erhardsson
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: torkel.erhardsson@liu.se
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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.

Keywords

Mixture of normal distributionsnormal approximationKolmogorov distanceStein’s methodphylogenetic treephenotypic traitYule processOrnstein–Uhlenbeck processjumps

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 92D15: Problems related to evolution
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 162 - 188
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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