The coalescent of a sample from a linear-fractional branching process

Area of Science:

• Stochastic processes
• Probability theory
• Population dynamics

Background:

• Bienaymé-Galton-Watson processes model population growth.
• Offspring distributions, specifically linear-fractional ones, are crucial for process behavior.
• Sampling individuals from a population is essential for data collection and analysis.

Purpose of the Study:

• To analyze genealogical trees of sampled individuals in Bienaymé-Galton-Watson processes.
• To compare tree distributions arising from Bernoulli sampling and uniform sampling.
• To establish a mathematical relationship between these two sampling schemes.

Main Methods:

• Focus on Bienaymé-Galton-Watson processes with linear-fractional offspring distributions.
• Implement Bernoulli sampling: each individual selected independently with a probability.
• Implement uniform sampling: a fixed number of individuals chosen uniformly at random.

Main Results:

• Genealogical trees were analyzed for both sampling methods.
• A direct relationship was established between the distributions of trees from Bernoulli and uniform sampling.
• The study quantifies how different sampling strategies impact the observed genealogical structure.

Conclusions:

• The choice of sampling method significantly influences the analysis of genealogical trees in population processes.
• Understanding the relationship between sampling schemes allows for more robust inferences.
• This research provides a theoretical foundation for analyzing sampled population data.