Probability

## Download Concentration inequalities and model selection: Ecole d'Ete by Professor Pascal Massart (auth.), Jean Picard (eds.) PDF

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By Professor Pascal Massart (auth.), Jean Picard (eds.)

Since the extraordinary works of Talagrand, focus inequalities were famous as basic instruments in different domain names akin to geometry of Banach areas or random combinatorics. in addition they develop into crucial instruments to enhance a non-asymptotic conception in information, precisely because the primary restrict theorem and big deviations are recognized to play a significant half within the asymptotic conception. an summary of a non-asymptotic conception for version choice is given right here and a few chosen purposes to variable choice, switch issues detection and statistical studying are mentioned. This quantity displays the content material of the path given through P. Massart in St. Flour in 2003. it's generally self-contained and available to graduate students.

Read or Download Concentration inequalities and model selection: Ecole d'Ete de Probabilites de Saint-Flour XXXIII - 2003 PDF

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Extra resources for Concentration inequalities and model selection: Ecole d'Ete de Probabilites de Saint-Flour XXXIII - 2003

Example text

Let t be given such that σt = E X 2 (t) > 0. 2 above and choosing z such that P [Z > z] ≤ 1/4, if we denote by Φ the cumulative distribution function of the standard normal distribution, we derive from the inequalities 3 ≤ P [Z ≤ z] ≤ P [X (t) ≤ z] = Φ 4 z σt , that σt ≤ z/Φ−1 (3/4). 1), it appears that the hard work to study the continuity question is to get dimension free bounds on the supremum of the components of a Gaussian random vector. One reason for focusing on supt∈T X (t) rather than supt∈T |X (t)| is that one can freely use the formula E sup (X (t) − Y ) = E sup X (t) t∈T t∈T for any centered random variable Y .

The product of a collection of Borel sets which are trivial except for a ﬁnite subcollection). The law of X is determined by the collection of all marginal distributions of the ﬁnite dimensional random vectors (X (t1 ) , . . , X (tN )) when {t1 , . . , tN } varies. , any stochastic process Y with same law as X is almost surely continuous on T (equipped with some given distance). s. for all t ∈ T . A (centered) Gaussian process is a stochastic process X = (X (t))t∈T , such that each ﬁnite linear combination αt X (t) is (centered) Gaussian (in other words, each ﬁnite dimensional random vector (X (t1 ) , .

42) implies that n Q [Xi = Yi ] ≤ minn Q∈P(P ,Q) i=1 n K (Q, P n ) 2 which is the original statement in [86]. On the other hand some other coupling result due to Marton (see [87]) ensures that n minn Q∈P(P ,Q) i=1 Ωn Q2 [Xi = Yi | Yi = yi ] dQ (y) ≤ 2K (Q, P n ) . 42) but with the suboptimal constant 2 instead of 1/2. Proof. We follow closely the presentation in [87]. 42) by induction on n. 20. Let us now assume that for any distribution Q on Ω n−1 , An−1 , P n−1 which is absolutely continuous with respect to P n−1 , the following coupling inequality holds true n−1 min n−1 Q∈P(P ,Q ) Q2 (x, y) ∈ Ω n−1 × Ω n−1 : xi = yi ≤ i=1 1 K Q , P n−1 .