how can one give an unbiased and more rigorous proof of the central limit theorem? in a series of papers over the past two decades, i have developed a new approach to the derivation of the central limit theorem and its generalizations using probability theory and optimal transport theory. in this talk, i describe this new approach and give examples of its use in the derivation of the central limit theorem, brownian motion, and the central limit theorem for continuous martingales.

this work is a significant extension of the author’s earlier work on the connection between infinite ergodic theory and infinite analysis. namely, the author gives new proofs of various basic results in the theory of general metric spaces, which in turn lead to new generalizations of results in both the theory of ergodic actions of countable groups and in the theory of probability measures on topological spaces. the proofs are based on a class of probability measures on metric spaces that are called gibbs measures and play a role similar to the role that probability measures play in ergodic theory.

this is a survey of the most recent advances in the theory of infinite ergodic theory. the general theme of the talk is that the theory has made a transition from being a relatively obscure tool for the study of markov chains to becoming a tool in many areas of mathematical physics, ergodic theory, and the analysis of probability measures.

the central limit theorem (clt) is a basic result in probability theory that underpins much of the statistical work in science and industry. in this talk, we give a new proof of the classical clt for independent and identically distributed (iid) random variables, which is significantly simpler than previous proofs. the proof applies to non-identically distributed random variables, and we generalize the result to martingales. we then describe an application of the martingale clt to the approximation of functionals of independent sums of martingale differences. 81555fee3f

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