Empirical Processes with a Bounded Ψ1 Diameter

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Nonparametric inference in generalized functional linear models (2014, methodology): “This would require a delicate analysis of the convergence of Vn, which may be handled by techniques in [18].”
MINIMAX SPARSE PRINCIPAL SUBSPACE ESTIMATION IN HIGH DIMENSIONS (2012, background): “The empirical process 〈Sn − ,UUT 〉 indexed by U is a generalized quadratic form, and a sharp bound of its supremum involves some recent advances in empirical process theory due to Mendelson (2010) and extensions of his results.”
Concentration inequalities and moment bounds for sample covariance operators (2014, methodology): “[11] Mendelson, S. (2010) Empirical Processes with a Bounded ψ1-Diameter.”
Upper bounds on product and multiplier empirical processes (2014, unknown): “t the probability estimate in [11] is far from optimal. Recently, Bednorz [2] obtained the optimal probability estimate under the assumption that F⊂ Lψ2 (deﬁned below), improving earlier results from [12, 8]. Deﬁnition 1.3 Let fbe deﬁned on the probability space (Ω,µ). For α≥ 1, the ψα norm of fis k”
Sparse PCA: Optimal rates and adaptive estimation (2012, methodology): “timating the principal subspace in term of the minimax rate. In the special case of r= 1, a similar combinatorial procedure has been proposed in [51]. Using Mendelson’s results on empirical processes [37], this procedure is shown to attain a convergence rate that is optimal in all parameters except ”