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Generalizations

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

Abstract

Concentration inequalities under log-Sobolev inequalities are optimal in the sense that they provide a Gaussian tail for statistics that are expected to satisfy a central limit theorem. However, for that very same reason, laws satisfying a log-Sobolev inequality must have a sub-Gaussian tail. One way to weaken this hypothesis is to weaken both requirements and hypotheses, for instance to assume a weaker coercivity inequality such as a Poincaré inequality. In this section, we keep the notations of the previous section. Let us recall that a probability measure ? on RM satisfies Poincaré’s inequality with coefficient m > 0 iff for any test function \({\rm f} \in C_b^2 (R^M )\)

$$\mu _\phi (\Gamma _1 (f,{\rm }f)) \ge m\mu _\phi [(f - \mu (f))^2 ].$$

Keywords

  • Riemannian Manifold
  • Bibliographical Note
  • Positive Ricci Curvature
  • Gaussian Tail
  • Compact Connected Manifold

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Alice Guionnet .

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© 2009 Springer-Verlag Berlin Heidelberg

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Guionnet, A. (2009). Generalizations. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_6

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