Sampling schemes of multitype continuous-time Bienaym\'e-Galton-Watson trees and limiting critical genealogies

Abstract

We study the genealogies of samples of $k$ distinguished particles drawn from the population alive at some fixed time in a continuous-time multitype Bienaym\'e-Galton-Watson (MBGW) process under two different type dependent sampling schemes: uniform sampling without replacement within types given a fixed type configuration, and sampling according to type-dependent weights. These schemes complement the uniform sampling at fixed time $T$ considered in Angtuncio, Pardo, C. Harris (2026a) which did not distinguish between sampled types. Under each scheme for a fixed sampling time $T$, we characterise the associated times of most recent common ancestors, ancestral offspring distributions, and type-dependent ancestral structure of the sample genealogy. In addition, under the assumption that the MBGW process is critical with finite second moments, we show that, conditional on survival of the population, a large time limiting sample genealogy emerges which is robust to the sampling scheme used. We identify this universal genealogy to have the same tree structure as the single-type case in C. Harris, Johnston, Roberts (2020), and we describe its ancestral type behaviour over scaled-times - this essentially being decoupled from the tree structure except at the times of ancestral splitting events.