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The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies $$\mathbb{P}\Big …

Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X_1,\ldots,X_n$ to be independent with the same distribution a …

Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix. What can be said about the distribution of the largest singul …

Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between …

In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables: http://terrytao.wordpress.com/2010/01/03/254a-no …