Approximation of the empirical distribution function

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    This series of problems discusses a result frequently used in probability theory and statistics, the optimal approximation of the standardized empirical distribution function by a Brownian bridge. This result and also the method of its proof is often called the KMT (Komlós, Major, Tusnády) method in the literature. The original proof of this result contains a rather concise proof, where the proof of several non-trivial details is omitted.

    This series of problems contains a detailed proof of the results. I also tried to explain the ideas behind the technical details. The paper contains a detailed discussion of some technical problems, because I hope that a detailed explanation of how to overcome certain technical difficulties may be interesting in itself. I mean in particular such technical details as the application of the Poissonian approximation to prove some ``self-evident'' estimates whose rigorous proof is not quite simple.

    48 pages