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An improvement of Tusnády’s inequality in the bulk. (English) Zbl 1482.60028

Strict bounds for the distance between Normal\(({m/2},{m/4})\) and Binomial\((m,1/2)\) distributions are provided by Tusnády’s inequality which “lies at the heart of the modern approach to the proof of the Komlós-Major-Tusnády approximation of the empirical process by a sequence of Bownian bridges”. The bounds are in terms of a standard \(N(0,1)\) distributed random variable \(Z\). The main result is a remarkable improvement of previous bounds if the values allowed for \(Z\) are restricted to \(|Z|\leq\sqrt{\log(m)}\) (the “bulk” of \(N(0,1)\)), where \(m\) is subject to usually large lower bound. The proof relies among other rather technical evaluations on a refined continuity correction for the Binomial distribution.

MSC:

60E15 Inequalities; stochastic orderings
62E17 Approximations to statistical distributions (nonasymptotic)
62E20 Asymptotic distribution theory in statistics
60F99 Limit theorems in probability theory
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References:

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