Tusnády’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian bridge. We present an elementary proof of a result that sharpens the Tusnády inequality, modulo constants. Our method uses the beta integral representation of Binomial tails, simple Taylor expansion and some novel bounds for the ratios of normal tail probabilities.
"Tusnády’s inequality revisited." Ann. Statist. 32 (6) 2731 - 2741, December 2004. https://doi.org/10.1214/009053604000000733