By Pascal Massart, Stéphane Boucheron, Gábor Lugosi

ISBN-10: 0199535256

ISBN-13: 9780199535255

Focus inequalities for features of self sufficient random variables is a space of likelihood idea that has witnessed an excellent revolution within the previous couple of a long time, and has purposes in a large choice of parts equivalent to desktop studying, statistics, discrete arithmetic, and high-dimensional geometry. approximately talking, if a functionality of many self sustaining random variables doesn't count an excessive amount of on any of the variables then it really is centred within the experience that with excessive chance, it truly is just about its anticipated price. This publication bargains a bunch of inequalities to demonstrate this wealthy idea in an available method by means of masking the major advancements and functions within the box.

The authors describe the interaction among the probabilistic constitution (independence) and various instruments starting from sensible inequalities to transportation arguments to details idea. functions to the examine of empirical procedures, random projections, random matrix idea, and threshold phenomena also are awarded.

A self-contained creation to focus inequalities, it incorporates a survey of focus of sums of self reliant random variables, variance bounds, the entropy process, and the transportation strategy. Deep connections with isoperimetric difficulties are printed when unique recognition is paid to functions to the supremum of empirical processes.

Written by way of prime specialists within the box and containing large workout sections this publication might be a useful source for researchers and graduate scholars in arithmetic, theoretical computing device technological know-how, and engineering.


The transparent exposition from uncomplicated fabric as much as contemporary refined effects and lucid writing sort make the textual content a excitement to learn. newcomers in addition to skilled scientists will prot both from it. it's going to definitely develop into one of many ordinary references within the box. Hilmar Mai, Zentralblatt Math

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Extra info for Concentration Inequalities: A Nonasymptotic Theory of Independence

Example text

Xi–1 , xi , Xi+1 , . . , Xn ) dμi (xi ). Then, again by Fubini’s theorem, Ei E(i) Z = Ei–1 Z. 1) This observation is key in the proof of the main result of this chapter which we state next. 1 (EFRON–STEIN INEQUALITY) Let X1 , . . , Xn be independent random variables and let Z = f (X) be a square-integrable function of X = (X1 , . . , Xn ). Then n Var (Z) ≤ E Z – E(i) Z 2 def = v. i=1 Moreover, if X1 , . . , Xn are independent copies of X1 , . . , Xn and if we define, for every i = 1, . .

Let b = sup S. Show that ψZ (λ) = log EeλZ is convex and infinitely many times differentiable on I = (0, b). Show that if EZ = 0, ψZ is continuously differentiable on [0, b) with ψZ (0) = ψZ (0) = 0 and the Cramér transform of Z equals ψZ∗ (t) = supλ∈I (λt – ψZ (λ)). 7. Prove that if Z is a centered normal random variable with variance σ 2 then t2 2σ 2 sup P {Z ≥ t} exp t>0 1 = . 8. (ELEMENTARY INEQUALITIES) Prove the following inequalities appearing in the text: – log(1 – u) – u ≤ u2 2(1 – u) h(u) = (1 + u) log(1 + u) – u ≥ h1 (u) = 1 + u – √ 1 + 2u ≥ for u ∈ (0, 1); u2 2(1 + u/3) u2 , 2(1 + u) for u > 0; for u > 0.

16 (MINKOWSKI’S INEQUALITY) Let X and Y be independent random variables taking their values in the sets X and Y, respectively. Let f : X × Y → R be a real-valued measurable function and define the random variable Z = f (X, Y). e. EX Z = E[Z|Y] and EY Z = E[Z|X]). Before proving the theorem, note that the classical version of Minkowski’s inequality cited above may be recovered by letting Y be uniformly distributed on the set Y = {1, 2} and defining X = (X1 , X2 ), f (X, 1) = X1 , and f (X, 2) = X2 .

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Concentration Inequalities: A Nonasymptotic Theory of Independence by Pascal Massart, Stéphane Boucheron, Gábor Lugosi

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