64 problems

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb E(X_i)=\mu_i\leq 1$ for each $i=1,\ldots,n$, and let $\delta>0$. Feige's conjecture. … This strengthe…

Talagrand's conjecture. For every fixed $\tau>0$, there is a constant $C_\tau$ such that, uniformly in the dimension and over all $f\geq 0$,

Let $L$ denote the set of all real random variables $X$ with a log-concave probability density function such that $EX=0$ and $EX^2=1$. Let $Y$ have the standard exponential distrib…

Conditional-regret conjecture. Behind all concentration results derived by constructing an appropriate non-negative (super)martingale and then applying Ville's inequality, there ex…

Let $U_n(r)$ be the union of the first $n$ sets selected by the cascading exclusion process at radius $r$, and let $W_n(r)$ denote the corresponding number of selected sets; let…

Small derivative-norm conjecture. The $L^2$-norms $\|\partial_M f\|_2$ should be small, particularly in dimensions $d\geq 3$, where exponential decay may occur.

Conjecture on improving $\alpha$. A smaller function $\alpha$ may be sufficient for the cited inequalities to hold, so that the resulting improvement could be carried over to the p…

Let $(X_i,\mathcal{F}_i)_{i\geq 0}$ be a supermartingale difference sequence, and let $x,y,v\geq 0$. Write $(u)_+=\max\{u,0\}$ and let $\widetilde{\Pi}_{v^2/y}$ denote the random v…

Donoho–Stark conjecture. Among all functions $f$ in the Paley–Wiener space and all measurable subsets $\Omega$ of the real line with fixed Lebesgue measure $|\Omega|=\ell$, the con…

Let $\nu=e^{-W}$ be a probability measure on $\mathbb{R}^d$, where $W$ is convex and $\|\operatorname{cov}(\nu)\|_{op}\leq 1$. An exponential concentration inequality with paramete…

Fixed-sample concentration conjecture. For a given sample size $n$, a concentration inequality of the same type should hold without the $\log(n)$ term.

Concentration conjecture. Under mild conditions, the probability that the number of real zeros deviates from its expectation by at least $\varepsilon\log n$ is bounded above and be…

Let $P=(p_j)_{j\geq 1}$ be a distribution, let $\mathbf{X}_n$ be an independent sample of size $n$, and let $Y_{n,j}$ denote the number of observations equal to the $j$-th type. Fo…

The paper studies zero-bias coupling and Wasserstein-bounding methods that produce efficient, computable concentration inequalities with explicit constants for certain statistics,…

Concentration conjecture. The irregular random subgraph $H$ satisfies

Let $G$ be a definable group in an o-minimal structure, let $S\subset G$ be a definable subset avoiding unboundedly large arithmetic progressions, and let $k=\dim(S)$. Consider the…

Exchangeable-set reduction conjecture. This restriction to exchangeable sets should hold not only in this special convexity case, but also in any other cases.

Concentration conjecture. There are positive constants $c_1$ and $c_2$ such that for every $n$, $k$, $P$, and $\varepsilon\geq 0$,

Leader–Radcliffe's conjecture.

Perpendicular-component concentration conjecture. A similar improvement should hold for the perpendicular component, replacing the $n^{-1/4}$ rate by

Perpendicular-component concentration conjecture. The deviation of $beta^+$ from its deterministic counterpart can likewise be improved to

Concentration-radius conjecture. This radius can be improved to order

Let $\boldsymbol\theta_k$, $\boldsymbol\theta_k^{\mathrm d}$, $\boldsymbol\theta_k^f$, and $\boldsymbol\theta_k^{f,\mathrm d}$ denote the original, debiased, fixed-design, and fixe…

Consider adaptive stopping procedures whose approximate type-I error guarantees are obtained from limit theorems, and alternative procedures using concentration inequalities to obt…

Let $C_n$ denote the facility-location objective for the random point set $X$, and let $r_p$ be the radius associated with each point $p\in X$. The displayed analysis applies Talag…