Title: Schnorr randomness and Levy's Upward Convergence Theorem
Speaker: Sean Walsh (UCLA)
Abstract: Much recent work in algorithmic randomness has concerned characterizations of randomness notions in terms of the almost-everywhere behavior of suitably effectivized versions of functions from analysis or probability. In this work, we consider Levy's Upward Martingale Convergence Theorem from this perspective. Generalizing a result of Pathak-Rojas-Simpson from Euclidean space to Cantor space, we show that Schnorr randoms are precisely the points at which the conditional expectations of L^1-computable functions converge to their true value, as one passes along sub-sigma-algebras generated by longer and longer strings. This result has natural applications to formal epistemology and the philosophical interpretation of probability: for, the natural Bayesian interpretation of this result is that belief, in the form of an agent's best estimates of the true value of a random variable, aligns with truth in the limit, under appropriate effecitivity assumptions. We also consider other convergence theorems and other randomness notions. No prior knowledge of algorithmic randomness will be assumed, and we will remind the audience of the various martingale convergence theorems which we are effectivizing. This is joint work with Simon Huttegger (UC Irvine) and Francesca Zaffora Blando (Stanford).