Abstract
Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.
Highlights
This article is a contribution to the functional approach to concentration inequalities, see, e.g., [19]
One obtains a dimension-free concentration inequality with a rate corresponding to the tails
For r ∈ (1, 2], we say that a probability measure μ on Rd satisfies the modified log-Sobolev inequality with parameter r if there exists a constant CmLS(r) < ∞ such that for every smooth function f : Rd → (0, ∞) one has
Summary
This article is a contribution to the functional approach to concentration inequalities, see, e.g., [19]. The Central Limit Theorem and an argument of Talagrand [23] roughly imply that if dimension-free concentration in Euclidean spaces occurs, the rate of concentration cannot be faster than Gaussian, and the measure should be exponentially integrable In this sense, Poincareand log-Sobolev inequalities describe the extreme dimension-free properties. For r ∈ (1, 2), Latała and Oleszkiewicz [18] showed that μr satisfies the inequality (5) with a uniformly bounded constant (in this case d = 1) For these measures, one obtains a dimension-free concentration inequality with a rate corresponding to the tails. For r ∈ (1, 2], we say that a probability measure μ on Rd satisfies the modified log-Sobolev inequality with parameter r if there exists a constant CmLS(r) < ∞ such that for every smooth function f : Rd → (0, ∞) one has. In view of Theorem 1, it is natural to conjecture, that the Latała–Oleszkiewicz inequality implies the modified log-Sobolev inequality (and improved two-level concentration), cf. Remark 21 in [5]
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