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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 …

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 …

Intuitively, it should approach zero fast as soon as $|\mathcal S_{\mathsf{big}}|$ is at all smaller than $n$ (even like $n/2$) because virtually all subsets $\mathcal S_{\mathsf{small}}$ will contain …

I wonder if this perhaps-naive approach can help you. Let's shift by the mean, so consider $X = \sum_{j=1}^n X_j$ where each $X_j = 0.5$ or $-0.5$ independently with one-half probability each. So $\ma …

Edit: I make the mistake below of proving a lower bound on the maximum number of boxes Alice must open, not the expected number. So this does not answer the question. . I think this sort of thing is …

I agree strongly with Peter in the comments: if you're interested in $p \gg \frac{\log n}{n}$, then don't do any hard work, just show the answer is essentially $p$ since the graph is essentially alway …

Suppose you plan to draw $n$ samples. Consider the following distribution: with probability $\frac{1}{2^n}$: draw from standard Cauchy with remaining probability: equal to zero This has infinite e …

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 …

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 …

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 …

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 …

For the expectation, linearity of expectation should help a lot. There are ${n\choose 2}$ pairs of chords, and if each chord is drawn i.i.d. then each pair has some probability of intersection of $p$, …

Intuition should say that if we keep tossing a die with a finite number of outcomes, then sooner or later we will see every outcome, in particular, we will see the maximum outcome. A proof is $$\Pr[\ …

This is a bit different and doesn't address the question, but hopefully close enough to be useful: we know some things about $\mathbb{E} L(Z,Y)$ for other loss functions $L$. If and only if $L$ is a …

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 …