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Multivariate Bernstein duality for tournament Wright--Fisher diffusions

  • Osvaldo Angtuncio Hernández
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Abstract

We give a probabilistic proof of a multivariate Bernstein duality for a multivariate Wright--Fisher frequency process with selection encoded by a finite tournament. The construction includes nontransitive interactions, such as rock--paper--scissors selection. Its dual is a pure-jump process of coefficient functions in the multinomial Bernstein basis. Selection increases the Bernstein degree through a winner--loser averaging operator, whereas neutral resampling decreases it through a duplication operator. For the proof, we construct a continuous forward process independently of duality, use duality to establish uniqueness, and obtain a weak stochastic differential equation representation and invariance of the simplex faces. This provides a sampling-based treatment of the tournament specialization of multitype Bernstein duality, extending an open problem in the binary Kingman setting of Cordero, Hummel, and Schertzer (2022).

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Cite

BibTeXRIS
BibTeX
@misc{hernandez2026multivariate,
  title = {{Multivariate Bernstein duality for tournament Wright--Fisher diffusions}},
  author = {Osvaldo Angtuncio Hernández},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.13731},
  eprint = {2611.13731},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.13731},
}
RIS
TY  - UNPB
TI  - Multivariate Bernstein duality for tournament Wright--Fisher diffusions
AU  - Osvaldo Angtuncio Hernández
PY  - 2026
DA  - 2026/11/10
PB  - arXiv
DO  - 10.48550/arxiv.2611.13731
UR  - https://arxiv.org/abs/2611.13731
AB  - We give a probabilistic proof of a multivariate Bernstein duality for a multivariate Wright--Fisher frequency process with selection encoded by a finite tournament. The construction includes nontransitive interactions, such as rock--paper--scissors selection. Its dual is a pure-jump process of coefficient functions in the multinomial Bernstein basis. Selection increases the Bernstein degree through a winner--loser averaging operator, whereas neutral resampling decreases it through a duplication operator. For the proof, we construct a continuous forward process independently of duality, use duality to establish uniqueness, and obtain a weak stochastic differential equation representation and invariance of the simplex faces. This provides a sampling-based treatment of the tournament specialization of multitype Bernstein duality, extending an open problem in the binary Kingman setting of Cordero, Hummel, and Schertzer (2022).
ER  -