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Maybe this is a stretch, but laws of large numbers, and more precise concentration of measure bounds, can be proven by the "isoperimetric" approach which can be called geometric and was pioneered by T …

The way this is usually put, the answer is that to achieve $\Pr[\|\hat{P}-P\|_1 \leq t] \geq 1- \delta$, one needs a sample size $N = \Theta\left(\frac{m + \ln(1/\delta)}{t^2}\right)$. The answer is t …

No, it looks like many different sufficiently symmetric distributions with enough concentration at $0$ will have $\arg\min_c E[|X-c|^p] = 0$ for all $p > 0$. Concrete family of examples: if $$ X = \be …

Don't forget that far out in the left tail, the Binomial CDF is multiplicatively approximated by the PMF, because terms grow geometrically. Example. For $t \leq \frac{np}{2}$, we claim $\Pr[X = t] \le …

Just an amateur answer, I would assume there's a paper or known approach out there from experts. I would partition the vertices into equal sets $U,V$ and delete edges to make it a bipartite graph. It …

The Good-Turing estimator addresses a very similar problem. The model is that there is an unknown number of types $n$ and a probability distribution $p \in \Delta_n$. Each time you pick up a new she …

Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta_n$. This question is about lower-bounding the max-loaded bin. Background. In this MO answer I …

Turns out that Gneiting and Raftery give an example in Section 4.2 of the continuous ranked probability score (CRPS), which is strictly proper for $\mathcal{P}$ equal to the Borel probability measures …

Well, it might be important to limit $\mathcal{P}$ here. If we consider the space $\Omega = \mathbb{R}$ with Lebesgue measure, we might take $\mathcal{P}$ to be the set of distributions with a continu …

Based on this cstheory post's argument from Clément Canonne[1], for $\delta = O(1)$ it suffices to draw $m = O(1/\epsilon^2)$ samples. This can be extended to show that $$ m = O\left(\frac{\log(1/\d …

A self-contained proof. Step 1: $\mathbb{E} \|\hat{P}_n - P\|_2^2 \leq \frac{1}{n}$. Step 2: McDiarmid's inequality. Let $X_i$ be the number of samples of $i \in \{1,\dots,k\}$. Then $X_i \sim \te …

Here's a sketch and a link for how I prove it. Let $$ f(x) = - \left( \frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5}\right) \phi(x) .$$ Now show (lemma): $\frac{df}{dx} = \left(1 + \frac{15}{x^6}\righ …

In a word, yes, KL is the only one. You're correct that $S$ is strictly proper if and only if $D_S$ is a Bregman divergence of some strictly convex function[1] (note you should swap the terms in your …

If you only want upper bounds, there is a nice approach based on collisions. (Proving that the max-loaded bin is not too small w.h.p. seems to need completely different techniques.) Define a $k$-way …

Adding to Iosif Pinelis' answer, there are two points here. First, as he says, the fact that we have a martingale rather than i.i.d. variables doesn't change much as proofs generally extend. So, secon …