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Talagrand's concentration inequality for empirical processes

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This is an expository talk about a deep probability inequality due to Talagrand (Invent. Math. 1996, cf. Ledoux’s book in 2001), which gives a Prohorov- (and then also Bernstein-) type exponential bound for the concentration of the supremum of an empirical process around its mean. I will try to discuss the following points: A) Some ideas of the proof, in particular the proof due to Ledoux using logarithmic Sobolev inequalities, which is related to the more general “concentration of measure” phenomenon. B) Discuss a variety of probabilistic applications, which should show how versatile this inequality is, in particular that it reproduces most known exp. inequalities for i.i.d. sums of (possibly Banach-)valued random variables. C) Discuss recent statistical applications to adaptive estimation, model selection problems, Rademacher processes, and almost sure limit laws (LIL-type results).

This talk is part of the Probability series.

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