Uniform Central Limit Theorems: Contents

By R. M. Dudley. 1999

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On the optimal estimation of probability measures in weak and strong topologies (2013, background): “cular, it is not easy to check whether there exists a pseudometric on FH such that FH is totally bounded (w.r.t. that pseudometric) and FH satisﬁes the asymptotic equicontinuity condition (see Dudley [10], Theorem 3.7.2) or FH satisﬁes the uniform entropy condition (see van der Vaart and Wellner [38”
Variance-based Regularization with Convex Objectives (2016, neutral): “sup x∈Xn N(F(x),ǫ,k·k∞), which bound the number of ℓ∞-balls of radius ǫrequired to cover F(x). Note that we always have N∞(F,ǫ,n) ≤ N(F,ǫ,k·kL∞(X)) by deﬁnition. The classical Dudley entropy integral [21, 47] shows that, if Pn denotes the point masses on x1,...,xn and k·kL2(P n) the empirical L2-nor”
Local Rademacher complexities (2005, methodology): “The proof uses the following result of Dudley [ 7 ], which relates the empirical Rademacher averages to the empirical L2 entropy of the class.”
Empirical margin distributions and bounding the generalization error of combined classifiers (2002, methodology): “See the discussion in the books of van der Vaart and Wellner (1996) and Dudley (1999) and references therein.]”
Nonparametric tests for conditional symmetry in dynamic models (2007, methodology): “Note that if P is compact, then !d reduces to the classical weak convergence concept of Hoffmann–Jørgensen (Dudley, 1999, p. 94).”
Nonparametric Instrumental Variable Estimation Under Monotonicity (2015, unknown): “5 in Dudley (2014) that φ(y)/Φ(y) ≥ (2/π), and so (50) implies that”
Global rates of convergence in log-concave density estimation (2014, background): “subsets A,B ! R d by Haus( A,B ):=max ! sup x " A inf y" B "x # y" , sup y" B inf x " A "x # y" " . Given any ! $ (0,!! 2 µ 1/2 2 (D )], by Lemma 8.4.3 of Dudley (1999), there exists c0 > 0such that µ 2 (D \ c D 0 !2 ) % ! 2 /4. Now, by the main result”
Program evaluation and causal inference with high-dimensional data (2013, background): “n of random variables F= ff(W;t);t2Tg, where f : W T !R, indexed by a set Tand viewed as functions of W2W, is suitably measurable with respect to Wif it is image admissible Suslin class, as dened in Dudley (1999, p 186). In particular, Fis suitably measurable if f: W T!R is measurable and Tis a Poli”
Combinatorics of random processes and sections of convex bodies (2004, background): “The precise definition can be found in [LT] and [Du 99].”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, background): “By Proposition 2.1 in Chernozhukov, Chetverikov, and Kato (2017), the probability on the right-hand side of (92) is bounded by P ( max 1≤j≤p Yj > (1− r)c0(βn + ϕ̄n) ) + Cn−c.”
Random Fields and Geometry (2007, methodology): “5), including extensions to Banach space valued processes for which ‖ ‖ is the norm, see the more detailed treatments of [29, 53, 64, 92, 98].”
On local U-statistic processes and the estimation of densities of functions of several sample variables (2007, methodology): “We refer to Dudley [10] or van der Vaart and Wellner [37] for the definition of Donsker classes of functions.”