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Let $p\in [0,1/2]$, and define $\xi$ as the symmetric random variable such that $$ \xi = \begin{cases} 1 & \text{ w.p. } p\\ 0 & \text{ w.p. } 1-2p\\ -1 & \text{ w.p. } p \end{cases} $$ so that $\math …

One quick remark: the upper bound will be $$ O\left(\min_j \frac{1}{H^2(p_j,q_j)}\right) $$ where $H$ is the Hellinger distance, not TV (we have $H^2\lesssim TV \lesssim H$). The sample complexity of …

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 …

In the course of a research project about discrete probability distributions, my coauthors and I keep seeing some quantity appear, and I would like to understand whether it has been studied or has a c …

I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X_n)_ …

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) …

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 have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\fra …

Let $n\geq 2$, and $X_1,\dots,X_n$ be independent non-central r.v.'s, where $X_i \sim \chi^2(\delta_i)$; and $w_1,\dots, w_n > 0$. Letting $$X \stackrel{\rm def}{=} \sum_{i=1}^n w_i X_i$$ is it true …

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions: The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > 0$ i …