Lower Bounds for Divergence in Central Limit Theorem Peter Harremo¨es 1 Department of Mathematics, University of Copenhagen Abstract A method for finding asymptotic lower bounds on information divergence is devel- oped and used to determine the rate of convergence in the Central Limit Theorem. Keywords: Central Limit Theorem, cumulant, Hermite polynomial, information divergence, kurtosis, maximum entropy, rate of convergence, skewness. 1 Introduction Recently Oliver Johnson and Andrew Barron [JB01] proved that the rate of convergence in the information theoretic Central Limit Theorem is upper bounded by c n under suitable conditions for some constant c. In general if r 0 > 2 is the smallest number such that the r’th moment does not vanish then a lower bound on total variation is c n r 0 2 −1 for some constant c. Using Pinsker’s inequality this gives a lower bound on information divergence of order 1 n r 0 −2 . In this paper more explicit lower bounds are computed. The idea is simple and follows general ideas related to the maximum entropy principle as described 1 Supported by a Post. Doc. fellowship by the Villum Kann Rasmussen Foundation and by grants from Danish Natural Science Counsil and INTAS (project 00-738). This work was mainly done during a stay at ZIF, Bielefeld. Electronic Notes in Discrete Mathematics 21 (2005) 309–313 1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2005.07.076