Seminario de Probabilidad y Procesos Estocásticos

Resumen: 

Exchangeability is an interesting property of random vectors with applications in probability theory and statistics. It often emerges in classical population genetics models, which study the transmission of types or traits in a population through time. In the classical Cannings model, a fixed-size and discrete generations model, individuals choose their parents from the previous generation in an exchangeable fashion. This ensures that the vector containing the types of the individuals stays exchangeable in every generation. In this work, we study the broader case of infinite population and identify the rules for the choice of ancestors that allow the exchangeability of the types vector to propagate forward in time. In particular, we provide examples of non-exchangeable forms of choice that still preserve types exchangeability from one generation to the next. De Finetti's theorem allows to derive random variables representing the frequency of a type in our infinite population, whose laws can be studied thanks to a moment duality result. Finally, our method sheds light on a nice countable representation of famous diffusions, such as the Wright-Fisher diffusion with selection (lookdown constructions).

https://www.matem.unam.mx/~seminarioproba/