The central limit theorem for the number of mutations in the genealogy of a sample from a large population

Yun-Xin Fu

Theoretical Population Biology, 2025, vol. 162, issue C, 22-28

Abstract: The number K of mutations identifiable in a sample of n sequences from a large population is one of the most important summary statistics in population genetics and is ubiquitous in the analysis of DNA sequence data. K can be expressed as the sum of n−1 independent geometric random variables. Consequently, its probability generating function was established long ago, yielding its well-known expectation and variance. However, the statistical properties of K is much less understood than those of the number of distinct alleles in a sample. This paper demonstrates that the central limit theorem holds for K, implying that K follows approximately a normal distribution when a large sample is drawn from a population evolving according to the Wright-Fisher model with a constant effective size, or according to the constant-in-state model, which allows population sizes to vary independently but bounded uniformly across different states of the coalescent process. Additionally, the skewness and kurtosis of K are derived, confirming that K has asymptotically the same skewness and kurtosis as a normal distribution. Furthermore, skewness converges at speed 1/ln(n) and while kurtosis at speed 1/ln(n). Despite the overall convergence speed to normality is relatively slow, the distribution of K for a modest sample size is already not too far from normality, therefore the asymptotic normality may be sufficient for certain applications when the sample size is large enough.

Keywords: Central Limit Theorem; Number of mutations; Asymptotic Normality; Skewness and Kurtosis; Coalescent Theory; Moment Generating Function; Speed of Convergence (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:thpobi:v:162:y:2025:i:c:p:22-28

DOI: 10.1016/j.tpb.2025.02.001

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