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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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.