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Register a new userProjections of probability distributions: A measure-theoretic Dvoretzky theorem
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Authors
Elizabeth Meckes
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Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on $R^d$, under mild conditions, most one-dimensional marginals are approximately Gaussian if $d$ is large. In earlier work, the author used entropy techniques and Steins method to show that this phenomenon persists in the bounded-Lipschitz distance for $k$-dimensional marginals of $d$-dimensional distributions, if $k=o(sqrt{log(d)})$. In this paper, a somewhat different approach is used to show that the phenomenon persists if $k<frac{2log(d)}{log(log(d))}$, and that this estimate is best possible.
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We present a novel proof of de Finettis Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finettis Theorem for standard Borel spaces is an instance of this result.
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