Another view of sequential sampling in the birth process with immigration

Area of Science:

• Stochastic processes
• Probability theory
• Mathematical biology

Background:

• The linear birth-immigration (BI) process is a fundamental model in population dynamics.
• Understanding family size distributions and their temporal correlations is crucial for ecological and epidemiological studies.
• Previous work has explored properties of the Chinese Restaurant Process (CRP), a related stochastic model.

Purpose of the Study:

• To investigate the correlation of family sizes observed in consecutive time intervals within a linear birth-immigration process.
• To analyze the expected sample variance and its asymptotic behavior for multiple sequential samples.
• To establish a connection between samples from a BI process and sequential samples from a CRP.

Main Methods:

• Utilizing a continuous-time framework for the linear birth-immigration process.
• Calculating expected sample variance and asymptotic properties for p consecutive samples.
• Conditioning on sample sizes to link BI process samples to CRP samples.
• Applying results to revisit Fisher's 1943 multi-sampling problem.

Main Results:

• Derived formulas for expected sample variance and its asymptotics in the BI process.
• Established a novel connection between sequential samples from a BI process and the CRP.
• Demonstrated that a continuous-time approach simplifies asymptotic calculations compared to discrete-time models.
• Provided a new interpretation of Fisher's multi-sampling problem within the context of sequential BI samples.

Conclusions:

• The study provides a comprehensive analysis of family size correlations in linear birth-immigration processes.
• The continuous-time framework offers computational advantages for asymptotic analysis.
• The findings offer new insights into stochastic modeling and have implications for understanding population dynamics and sequential sampling problems.