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This paper is concerned with the information-theoretical limits of density estimation for Gaussian random variables with data drawn independently and with identical distributions. We apply Fano's inequality to the space of densities and an arbitrary estimator. We derive necessary conditions on the sample size for reliable density recovery and for reliable density estimation. These conditions are true simultaneously for both finitely and infinitely dimensional density spaces.
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James A. Brofos, "Information-Theoretic Limits for Density Estimation." Dartmouth Computer Science Technical Report TR2014-765, December 2014.
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