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This shouldn't be possible without assumptions on $X$. Take any algorithm drawing $m$ samples. Construct a discrete distribution $X$ by drawing $2^m$ samples indepedently and uniformly from $[0,1]$, a …

Just saw this question. As KEW says in the comments, questions like these are studied in theoretical CS as it intersects with stats. It depends on the measure used to define "good". One natural choic …

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[\ …

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 …

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 …

In general, you are just asking about a weighted sum of i.i.d. variables from distribution $D$, with weights $\alpha_1,\dots,\alpha_n$. The Gaussian distribution is the only one that is rotationally i …

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 …

I think this is a natural question but am not sure where to find resources. Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one su …

Main claim. $$\Pr[|X - \mathbb{E} X| \geq t] \leq \frac{\mathbb{E} X}{t^2} \approx \frac{L}{t^2}. $$ You can bound the variance and use Chebyshev's Inequality as you suggest, and the calculations are …

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 …