A random variable is `concentrated' if it stays close to a fixed constant with high probability. The most basic examples are long averages of i.i.d. random variables, whose concentration is promised by the law of large numbers. However, concentration occurs in many other settings, even when exact calculations of moments or other parameters are impossible. This is because concentration often results from some natural `geometric' structure on the underlying probability space. For instance, large classes of random variables exhibit concentration because they are Lipschitz functions on certain `high-dimensional' spaces that carry both metrics and probability measures, such as Hamming cubes or high-dimensional spheres.
Lecture 3 will introduce some more abstract applications of the concentration of measure phenomenon in ergodic theory. Here our interest turns from particular random variables to the geometry of certain metric probability spaces `as a whole'. We focus on certain sequences of such spaces which arise naturally in the study of measure-preserving systems, and discuss when these spaces exhibit concentration and what the consequences are. Pre-requisites: same as Lecture 1, /plus/ some material on sofic entropy from Lewis Bowen's tutorials.
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