Approximation of partial sums of independent random variales

This series of problems is not finished. It contains the formulation of the problems and the proof of those results which are the most important in the proof and whose proof is only sketched in the original paper.

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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