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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 …

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 …

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 …

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)_ …

Many property testing algorithms in the dense graph model rely on the Szemerédi regularity lemma [Sze78], which (essentially) guarantees that every large enough graph can be divided into parts of roug …

Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the average distance of $S$ as $$ \bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum_{x,y\in S} d_H(x,y)\tag{1} $$ where …

My favorite proof so far is the one from Theorem 4.5 of Yihong Wu's lecture notes, which uses the data processing inequality to reduce the problem to the binary case: if $S$ is a measurable subset of …