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

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  • Fu, Yun-Xin

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.

Suggested Citation

  • Fu, Yun-Xin, 2025. "The central limit theorem for the number of mutations in the genealogy of a sample from a large population," Theoretical Population Biology, Elsevier, vol. 162(C), pages 22-28.
  • Handle: RePEc:eee:thpobi:v:162:y:2025:i:c:p:22-28
    DOI: 10.1016/j.tpb.2025.02.001
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