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Suppose $p \leq q \leq 1/2$, and $n,m\geq 1$ two integers. Let $X\sim \mathrm{Bin}(n,p)$, $Y\sim \mathrm{Bin}(m,p)$ and $X'\sim \mathrm{Bin}(n,q)$, $Y'\sim \mathrm{Bin}(m,q)$ be independent. Is it tr …

The conjecture is false, as shown by mschauer. (At least if we allow $m\neq n$; the case $m=n$ is still unclear to me.) Below is the simplest counterexample I could find in hindsight. Following mscha …

Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}\left[ \exp\ …

. $$ The inequalities established in a roundabout way: The relation between the two quantities shown by combining [1] and [2] (which both establish sample complexity upper and lower bounds for the same …

Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$, $$ H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n) …

I would have a proof of the following fact; but it's a bit clunky, and am wondering if one can get a more elegant one (and/or improve the constants). I couldn't find this anywhere, and searching prope …

I have an indirect (but simple) proof, which relies on the chi-squared distance $\chi^2(p,q) = \sum_n \frac{(p(n)-q(n))^2}{q(n)}$ as a proxy, along with the fact that $H(p,q)^2 \leq 1\land \chi^2(p,q) …

Not sure how interesting it is, given that computing $\mathbb{E}[|X-\mathbb{E}[X]|]$ may be unwiedly, but Iosif Pinelis' argument can be adapted to give the following statement, which does not require …

Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$. It is not hard to …