Abstract
When two (possibly different in distribution) continuous-state branching processes with immigration are present, we study the relative frequency of one of them when the total mass is forced to be constant at a dense set of times. This leads to a SDE whose unique strong solution will be the definition of a Λ-asymmetric frequency process (Λ-AFP). We prove that it is a Feller process and we calculate a large population limit when the total mass tends to infinity. This allows us to study the fluctuations of the process around its deterministic limit. Furthermore, we find conditions for the Λ-AFP to have a moment dual. The dual can be interpreted in terms of selection, (coordinated) mutation, pairwise branching (efficiency), coalescence, and a novel component that comes from the asymmetry between the reproduction mechanisms. In the particular case of a pair of equally distributed continuous-state branching processes the associated Λ-AFP will be the dual of a Λ-coalescent. The map that sends each continuous-state branching process to its associated Λ-coalescent (according to the former procedure) is a homeomorphism between metric spaces.
Funding Statement
The second author was supported by the grant CONACYT CIENCIA BÁSICA A1-S-14615.
Acknowledgments
We want to thank the anonymous referees for the careful reading, constructive comments, and suggestions, which significantly improved the presentation and the readability of the paper. Specially, for the detailed advice provided by one of the referees to shorten the proof of Theorem 3. Adrián González Casanova is a Neyman Visiting Professor at Department of Statistics of the University of California, Berkeley.
Citation
Maria Emilia Caballero. Adrián González Casanova. José-Luis Pérez. "The relative frequency between two continuous-state branching processes with immigration and their genealogy." Ann. Appl. Probab. 34 (1B) 1271 - 1318, February 2024. https://doi.org/10.1214/23-AAP1991
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