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Fleshing out Boris Bukh's idea. We can draw $\pi$ by first sending $1$ uniformly to somewhere in $\{1,\dots,n\}$, then sending $2$ uniformly to the remaining $n-1$ spots, and so on. Consider a small …

Is there a convex function $F$ that is not differentiable, but whose subdifferential admits a continuous selection, i.e. a continuous $g$ with $g(x) \in \partial F(x)$ for all $x$ in the domain? In on …

Here's a proof I like but with an exercise left in it for the reader. Let $c_i = |x_i-y_i|$, let $A = \{i : x_i \geq y_i\}$ and $B$ the remainders, and let $c = \sum_{i \in A} c_i = \sum_{i \in B} c_i …

One answer - subgaussian variables generalize this property. Let $\mu = \sum_i p_i z_i$, then the distribution is considered $\sigma^2$-subgaussian if for all $\lambda \in \mathbb{R}$, $$ \log\left …

As you said, the sum is $\Pr[X \leq \alpha n]$ where $X$ is drawn from a Binomial distribution with $n$ trials having $p$ probability of success. Bounds on this sum (for $\alpha < p$) are called "tail …