Entropic CLT for Order Statistics

TL;DR

This paper proves an entropic version of the Central Limit Theorem for order statistics, showing faster convergence rates under mild conditions, with potential implications for statistical theory.

Contribution

It establishes an entropic CLT for order statistics with an explicit convergence rate, strengthening the classical CLT results using relative entropy.

Findings

Entropic CLT for order statistics with O(1/√n) convergence rate

Derived ancillary results on order statistics of independent samples

Provides conditions under which the entropic convergence holds

Abstract

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 CLT that ensures a stronger mode of convergence using the relative entropy. In particular, an order $O(1/\sqrt{n})$ rate of convergence is established under mild conditions on the parent distribution of the sample generating the order statistics. To prove this result, ancillary results on order statistics are derived, which might be of independent interest.