Limit Theorems for $U$-Processes

By Miguel A. Arcones, Evarist Gin'e. 1993

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Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean (2004, background): “Indeed, as seen in Example 4.4, J ( F)(x0) is then a non-positive random variable, and hence is nonGaussian.”
High-dimensional consistent independence testing with maxima of rank correlations (2018, methodology): “We use the Hoeffding decomposition and the exponential inequality for bounded completely degenerate U-statistics of Arcones and Giné (1993) to prove the general case m ≥ 2.”
Rademacher Chaos Complexities for Learning the Kernel Problem (2010, background): “gif" alt-version="no" mimetype="image" xlink:type="simple"> One can refer to Arcones and Gine ( 1993 ) and De La Pena and Gine ( 1999 ) for more general entropy integrals to bound the suprema of Rademacher chaos processes of order m for any <inline-graphic xlink:href="NECO_a_00028inline184.”
Expected‐posterior prior distributions for model selection (2002, unknown): “…distribution F when E F f ( Y l 1 ; Y l 2 ; : : : ; Y lm j ) m N ( Y l 1 ; Y l 2 ; : : : ; Y lm ) < 1 : Uniform convergence results can also be obtained by noting that ( ) == N ( ) is a U - process indexed on f ( j ) =m N ( ) , for 2 ( Nolan and Pollard , 1987 , 1988 ; Arcones and Gin e , 1993 ) .”
Structural properties and convergence results for contours of sample statistical depth functions (2000, background): “Results on (B.1) have been established for the halfspace depth by Donoho and Gasko (1992), the simplicial depth by Liu (1990), Dümbgen (1990) and Arcones and Giné (1993), and the majority and Mahalanobis depths by Liu and Singh (1993).”
Power Transformation Toward a Linear Regression Quantile (2007, background): “11 of Arcones and Giné (1993) gives supλ∈ λτ |V0 n (λ) − V(λ)| = op(1). The term In on the right side of (A.14) can be shown to be op(1) by applying lemma 2.8 of Pakes and Pollard (1989). The consistency of β̂n follows from the uniform consistency of β̂(λ), as established in lemma 4 of Mu (2005).”
Jackknife multiplier bootstrap: finite sample approximations to the U-process supremum with applications (2017, background): “As a significant departure from the existing literature [27, 4, 14, 15], our Gaussian approximation for U -processes has a multi-resolution nature, which is neither parallel with the theory of U -processes with fixed function classes nor that of empirical processes.”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, background): “When p (and therefore d) is fixed, the classical central limit theorems (CLT) for approximating Tn by a Gaussian random vector T \ n ∼ N(0,Γ), where Γ = Cov(g(X1)) and g(X1) = E[h(X1, X2)|X1] − θ, have been extensively studied in literature [36, 3, 39, 33, 26, 37, 34, 67, 3, 75, 32, 38, 39, 67].”
Empirical process results for exchangeable arrays (2019, background): “As a result, the inequality proved on page 1508 in Arcones and Giné (1993) is valid with their f replaced with h (in particular, the sixth inequality on the latter page is true as h is symmetric in its arguments and h(·) does not depend on i) and we can write for some constant Ck that depends on k…”
General notions of statistical depth function (2000, unknown): “…halfspace depth, HDn x = inf P̂n H H a closed halfspace x ∈H x ∈ Rd where P̂n denotes the usual empirical measure, and Liu (1990), Dümbgen (1990), and Arcones and Giné (1993) for the sample simplicial depth SDn x = ( n d+ 1 )−1 ∑ 1≤i1 ··· id+1≤n I x ∈ S Xi1 Xid+1 x ∈ Rd For the sample…”
Random sampling of 3-colorings in Z 2 (2004, unknown): “Some other large deviation bounds for U-statistics are given in [1], [ 2 ], [7].”