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Metric entropy of high dimensional distributions. (English) Zbl 1147.46018

For a collection \(T\) of probability distributions in \([0, 1]^d\) having a density, the authors give estimates for the metric entropy under the \(L^2\)-norm. They obtain its exact rate for \(d=1\) and \(d=2\) and get bounds for \(d > 3\). Then, they establish connections to the small deviation probability for Brownian sheets under the sup-norm.

MSC:

46B50 Compactness in Banach (or normed) spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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