Approximation of the empirical distribution function
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Series of problems in Probability Theory
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.