Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities

By Sergey G. Bobkov, Friedrich Götze. 1999

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Transportation-information inequalities for Markov processes (2007, methodology): “[5] S.G. Bobkov and F. Götze.”
A large deviation approach to some transportation cost inequalities (2005, result): “As regards item (iii), the main technical (easy) result is Theorem 3.7 which is an extension of Bobkov and Götze’s characterization of T1(C) stated at (1.4).”
Ricci curvature for metric-measure spaces via optimal transport (2004, background): “Again in the case of Riemannian manifolds, a further circle of ideas relates displacement convexity to log Sobolev inequalities, Poincaré inequalites, Talagrand inequalities and concentration of measure [7] [8] [25] [33].”
CONCENTRATION OF THE SPECTRAL MEASURE FOR LARGE MATRICES (2000, background): “Recall also that the Gaussian law [12], any probability measure absolutely continuous with respect to Lebesgue measure satisfying the Bobkov and G¨ otze [ 6 ] condition (including (dx )= Z 1ej xj dx for 2), as well as any distribution absolutely continuous with respect to them possessing a bounded a”
Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance (2011, background): “By Chebyshev’s bound, for θ > 0, P( 1 n ∑ f(Xi)− ∫ fµ ≥ ε) ≤ exp−n[θε− log( ∫ eθf(x)µ(dx)e−θ ∫ fµ)] According to Bobkov-Götze’s dual characterization of T1, the term inside the log is bounded above by eCθ 2 , for some positive C, whence P( 1n ∑ f(Xi)− ∫ fµ ≥ ε) ≤ exp−n[θε− Cθ2].”
On the role of convexity in functional and isoperimetric inequalities (2008, background): “In this section, we introduce a coherent unified framework that provides an equivalence between capacity inequalities and weak-type Orlicz–Sobolev functional inequalities (introduced below), and a general sufficient condition for an equivalence to Orlicz–Sobolev inequalities.”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, background): “The arguments of [8] rely on transportation inequalities such as the celebrated Bobkov-Götze inequality [5].”
Probability distance inequalities on Riemannian manifolds and path spaces (2004, unknown): “Based on an argument of [5] for the transportation cost inequality corresponding to the log-Sobolev inequality, we have the following concentration result for (1.”
Bulk universality for generalized Wigner matrices (2010, methodology): “Using the Bobkov-Götze concentration inequality [4] and the uniform bound on the LSI constant (6.”
Kantorovich duality for general transport costs and applications (2014, background): “Although transport inequalities involving W 2 1 instead of W 2 2 (the so called T1 inequalities) make perfect sense in discrete settings (see Bobkov-Götze [8], Djelout-Guillin-Wu [19], Bobkov-Houdré-Tetali [9]), these inequalities tensorize only with a constant depending on the dimension! So the W1 ”