Topics in high-dimensional probability

[27.09 Slightly updated version of the sheet]

Please take a look at Exercise 2 of this sheet, which will provide you with an answer regarding variance of the maximum of a Gaussian vector. 

Remark: since one proves Gaussian Poincare inequality using Efron-Stein, of course, the latter will also give you the desired bound, however, only through a highly non-trivial application of it (recall that in the proof you have used additional Rademacher variables and CLT), but not through a natural application of the inequality.

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