Algebra Seminar talk
2024-06-21
David Schrittesser
Generalizing de Finetti -- WEGEN KRANKHEIT ABGESAGT
Abstract:
(SORRY, THE TALK FOR JUNE 21 2024 WAS CANCELLED!)
Intuitively, de Finetti's theorem states that if we make a sequence of measurements in a setting where we know it to be irrelevant in which order the measurements are obtained, then these measurements are conditionally independent, that is, independent given some latent random element. To be more precise, here is one version of de Finetti's theorem: Given a sequence of real random variables X1, X2, ... whose joint distribution is invariant under permutations of the indices, if we condition each Xi on the exchangeable algebra E obtaining the random variable (Xi | E), then the (Xi | E) are identically and independently distributed.
It turns out that the assumption that the state space is "nice" (here, the real numbers) is crucial to de Finetti’s. One can ask if de Finetti's theorem holds for sequences of random elements whose state space is some more general measure space (that is, not just for sequences of real random variables). In this talk, I discuss this question. In particular, I give a characterization of conditionally identically independent sequences without any assumptions on the state space, as well as a version of de Finetti for sequences whose common distribution is Radon (strengthening theorems due to Irfan and Fremlin).
This is joint work with Peter Potaptchik (Oxford) and Daniel M. Roy (University of Toronto).