Decoupling From Dependence To Independence

By M. Frei. 2016

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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].”
Risk minimization by median-of-means tournaments (2016, methodology): “Indeed, if f − f ∗ L q ≤ L f − f ∗ L 2 , then by the Paley-Zygmund inequality (see, e.g., [12]), for constants κ 0 and ρ 0 that depend only on L and q .”
Minimax rates for memory-bounded sparse linear regression (2015, unknown): “Letting ri ∈ {±1} be i.i.d. Rademacher variables and Vi = ( (i))2 be shorthand, an immediate symmetrization inequality (e.g. de la Peña and Giné, 1999, Lemma 1.2.6) gives that P [ n∑ i=1 Vi ≥ σ2n+ t ] ≤ P [ n∑ i=1 Vi ≥ n∑ i=1 E [Vi] + t ] ≤ t−pE [∣∣∣∣∣ n∑ i=1 (Vi − E [Vi]) ∣∣∣∣∣ p] ≤ 2pt−pE…”
Minimax optimality of permutation tests (2020, methodology): “We are then able to apply concentration inequalities for degenerate U -statistics in De la Pena and Giné (1999) to finish the proof.”