The coalescent structure of multitype continuous-time Bienaymé-Galton-Watson trees
Juan Carlos Pardo Osvaldo Angtuncio Hernández Simon Harris
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Abstract
We study the genealogy of a sample of $k>1$ particles, taken under various sampling schemes, from the population alive at fixed times, in a continuous-time multitype Bienaymé-Galton-Watson (MBGW) tree with finite second moments. For critical MBGW trees under uniform sampling without replacement, we show that the sample genealogy converges in the large time limit to a universal limiting structure with the same tree topology as Kingman's coalescent but enriched by a nontrivial process coding the types along each lineage. Remarkably, the limiting genealogy is robust to the sampling scheme: it remains unchanged when fixing type configurations (that is, if we sample uniformly at random without replacement given a fixed vector of types to sample), or sampling with type-dependent weights (that is, multinomial sampling over types). Tracking types along the genealogy reveals that, in the limit, types evolve independently of the tree structure but retain a distribution determined by the offspring law, and the Perron-Frobenius eigenvalues. For uniform sampling without replacement, we use $k$ distinguished \emph{spine} particles and a suitable change of measure under which, when sampling at fixed times: (a) the spines form a uniform sample without replacement that depend on the types, and (b) there is $k$-size biasing and discounting according to the population size. This work substantially extends the spine techniques developed by Harris, Johnston, and Roberts 2020 and Harris, Johnston, and Pardo 2024, for the single-type case. In particular, we provide a detailed analysis of how type information affects functionals of the spines. We show that the multitype case introduces complex interactions between types, resulting in a richer dependency structure where functionals must capture type-specific behaviours and inter-type correlations.引用本文复制引用
Juan Carlos Pardo,Osvaldo Angtuncio Hernández,Simon Harris.The coalescent structure of multitype continuous-time Bienaymé-Galton-Watson trees[EB/OL].(2026-02-24)[2026-02-27].https://arxiv.org/abs/2502.04588.学科分类
数学
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