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Michael Casey : Introduction to Concentration of Measure

The weak law of large numbers states that for a sequence of independent identically distributed random variables of finite mean, their average converges to this mean in probability as the number of terms tends to infinity. Well, how fast? That is, how many draws must we make before we see this behavior? For many distributions, the convergence is exponentially fast in the number of terms. Such behavior is a hallmark of concentration of measure: under suitable conditions, well behaved functions of many random variables do not deviate much from a particular value. In this talk, we'll show that such properties are not mysterious, but can be derived from a simple recipe using a few choice inequalities. Examples in both the discrete & continuous settings will be given, making connections with convex geometry.

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