Tusnady's lemma, 24 years later
As for the coupling between a binomial random variable \(B_n\) with parameter \((n,1/2)\) and a normal random variable \(Y\) the author establishes the following inequality: \[ \left|B_n-{n\over 2}-{\sqrt n\over 2}Y\right |\leq {3\over 4}+ {Y^2\over 8}. \] The constant \(\frac 34\) in the right hand side of the above inequality improves on the constant 1 appearing in Tusnády's original inequality [cf. \textit{J. Bretagnolle} and \textit{P. Massart}, Ann. Probab, 17, No. 1, 239-256 (1989; Zbl 0667.60042)].
scientific article; zbMATH DE number 1879945