In the mini-course to come, we will go through lots of concentration inequalities. There is a unite way to think about them which I will try to explain in this post.

Consider a sequence of i.i.d. random variables Consider their empirical mean

The interesting point about it is that for any (so, in a sense, the scaling is natural), but *concentrates* above its mean:

In fact, as we will see in the next post,

i.e. with exponentially high probability.

This is the simplest example of the **concentration of measure** phenomenon*,* when a function which is weakly dependent on many independent random variables concentrates around its mean or median. A collateral goal of the course is to observe similar properties in different settings: with broader classes of functions (Lipschitz functions, norms, quadratic forms, etc.), distributions of and mutual dependency in the sequence e.g. martingales.

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## About Dmitry Ostrovsky

PhD student with interests in statistics, optimization, and machine learning.

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