Entropic Central Limit Theorem for Order Statistics

Martina Cardone, Alex Dytso, Cynthia Rush

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


It is well known that central order statistics exhibit a central limit behavior and converge to a Gaussian distribution as the sample size grows. This paper strengthens this known result by establishing an entropic version of the central limit theorem that ensures a stronger mode of convergence using the relative entropy. This upgrade in convergence is shown at the expense of extra regularity conditions, which can be considered as mild. To prove this result, ancillary results on order statistics are derived, which might be of independent interest. For instance, a rather general bound on the moments of order statistics, and an upper bound on the mean squared error of estimating the p (0,1) -th quantile of an unknown cumulative distribution function, are derived. Finally, a discussion on the necessity of the derived conditions for convergence and on the rate of convergence and monotonicity of the relative entropy is provided.

Original languageEnglish (US)
Pages (from-to)2193-2205
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - Apr 1 2023
Externally publishedYes

Bibliographical note

Funding Information:
The authors would like to thank the Associate Editor and the Reviewers for their suggestions and for a speedy review process.

Publisher Copyright:


  • Central limit theorem
  • median
  • order statistics
  • quantiles
  • relative entropy


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