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.
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Research partially supported by the National Science Centre, Poland, grants no. 2015/19/N/ST1/00891 (MS) and 2017/24/T/ST1/00323 (doctoral scholarship of MS).
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Barthe, F., Strzelecki, M. Functional Inequalities for Two-Level Concentration. Potential Anal 56, 669–696 (2022). https://doi.org/10.1007/s11118-021-09900-9
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DOI: https://doi.org/10.1007/s11118-021-09900-9