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.

已知满足 Poincaré 不等式的概率度量享有指数率的无量纲集中不等式。Bobkov 和 Ledoux 的一个著名结果表明，庞加莱不等式自动隐含了修改后的对数 Sobolev 不等式。因此，庞加莱不等式确保了更强的无量纲集中属性，称为两级集中。我们表明 Latała-Oleszkiewicz 不等式发生了类似的现象，该不等式旨在揭示指数和高斯之间的速率的无量纲集中。在寻找相关问题的反例的推动下，我们还开发了分析技术来研究具有野生潜力的概率测度的函数不等式。