Radek Adamczak: Functional inequalities and concentration of measure I
Автор: Hausdorff Center for Mathematics
Загружено: 2021-01-11
Просмотров: 805
Описание: Concentration inequalities are one of the basic tools of probability and asymptotic geo- metric analysis, underlying the proofs of limit theorems and existential results in high dimensions. Original arguments leading to concentration estimates were based on isoperimetric inequalities, which are usually difficult to obtain. Over the years however simpler methods were found, allowing for exten- sion of concentration inequalities to more general classes of measures. In the lectures I will discuss the arguably softest approach to concentration, relying on functional inequalities. The focus will be put on classical inequalities, such as the Poincare inequality, the log-Sobolev inequality and some of their modifications. I will present their basic common properties (e.g., tensorization) and show how they imply various forms of concentration for Lipschitz functions. I will first present the continuous setting, starting with the simplest cases of the exponential and Gaussian measures, and subsequently move to discrete examples. If time permits I will also discuss some concentration results for non-Lipschitz functions, which can be obtained from functional inequalities.
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