On the genealogy of large populations

Key Points

• To introduce a mathematical Markov chain framework that models genealogical relationships among individuals sampled from a generation in a large haploid population.
• Formulated a new Markov chain to trace common ancestry among n sampled individuals.
• Utilized probabilistic coupling techniques to evaluate the mathematical properties of the process simultaneously across all sample sizes n.
• Established a rigorous mathematical representation of lineage branching and coalescence for arbitrary sample sizes.
• Showed that existing theoretical results in neutral mutation theory arise as direct consequences of the proposed genealogical chain.