Tusnady’s Lemma, 24 Years Later

– The optimal coupling between a variable with the Bin(n,1/2) distribution and a normal random variable lies at the heart of the proof of the KMT Theorem for the empirical distribution function. Tusnády’s Lemma (published in 1977 in his dissertation and in Hungarian) provides an inequality with explicit absolute constants which says that for this coupling, the distance between the random variables remains bounded in probability. In the appendix of a joint work with Jean Bretagnolle (1989), we have proposed a proof of Tusnády’s Lemma which though elementary is highly technical and considered as rather obscure, at least this is what we have understood from several conversations with motivated readers. The purpose of this paper is to provide an alternative proof which is still based on elementary computations but which we hope to be simpler and more illuminating. This new proof also leads to a slight improvement on the original result in terms of constants.  2002 Éditions scientifiques et médicales Elsevier SAS MSC: primary 60F17, 60F99; secondary 62G30