32 problems

Let $q>0$ be real, let $a_1,\dots,a_n$ be non-negative real numbers satisfying … and let $X_1,\dots,X_n$ be independent identically distributed standard exponential random variable…

Let $\boldsymbol{Z}=(Z_1,\ldots,Z_d)$ be a centered Gaussian random vector with $d\in\mathbb{N}$, and let $m\in\mathbb{N}_0$. Gaussian product inequality. One has … The conjecture…

Latała's moment conjecture. There exists a universal constant $C$ such that, for every log-concave vector $X$ with values in a finite-dimensional normed space $(F,\|\ \|)$ and ever…

The properties CNA+, JNRD+ and h-NLC+ are stronger versions of CNA, JNRD and hereditary NLC, respectively, requiring the corresponding property after every imposition of an externa…

Higher-even-moment collapse conjecture. For the classes $\mathbb{E}X^{2m}\leq\kappa_{2m}$, the one-sided/two-sided collapse of the cited collapse result persists beyond the corresp…

Gaussian correlation conjecture. One has

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…

Let $(X_k)$ be independent copies of a nondegenerate nonnegative random variable, let $S_n=X_1+\cdots+X_n$, and set $Z_n=\sum_{k=1}^n e^{\imath S_k}$. For $p>0$, write…

Let $X\sim f_X$ and $Z\sim f_Z$ be independent random variables, where $f_X$ and $f_Z$ are densities with cumulative distribution functions $F_X$ and $F_Z$, respectively. Integral…

Failure conjecture. The condition

Let $G$ be a graph. A function $f:G\times\{0,1\}^{E(G)}\to\mathbb{R}$ is a monotone cluster property if it depends only on the cluster of its vertex, is increasing under enlargemen…

Let $G$ be a graph, let $A\subset G$, and let $0,b\in G$ be vertices. The uniform continuity conjecture. For every $\varepsilon>0$ there exists $\delta>0$ such that, whenever … and…

Let $G$ be a finite graph with arbitrary probabilities on its edges. Let $0,b\in G$ be vertices and let $A\subset G$ be a set of vertices. The pre-FKG conjecture. … This is a forma…

Let $G$ be a finite graph with arbitrary probabilities on its edges. Let $0,b\in G$ be vertices and let $A\subset G$ be a set of vertices. The post-FKG conjecture. … This is an FKG…

Let $Z_1,Z_2,\ldots,Z_n$ be independent and identically distributed random variables, with each $Z_i$ representing a cluster of $m>1$ observations, and suppose that $c(i)=1$ for al…

Let $p\in\mathbb{N}$, let $\alpha\in(p-1,\infty)$, and let $\Sigma\in\mathcal{S}_{++}^p$. Let $\mathfrak{X}\sim\mathcal{W}_p(\alpha,\Sigma)$ be a $p\times p$ Wishart random matrix,…

Let $\boldsymbol{Z}=(Z_1,\ldots,Z_d)$ be the Gaussian vector in the strong Gaussian product inequality, with $d\in\mathbb{N}$, and let…

Let $U_1,U_2,\ldots$ be independent random variables uniformly distributed on $[-1,1]$, let $p\in(0,1)$), and let $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$. The sharp $L_p$-$L_2$ moment…

Lewicki–Skrzypek conjecture. For every $n\in\mathbb{N}$,

Let $X$ have a Poisson binomial distribution $Q$ generated by probabilities $p_i\in[0,1)$, let $\lambda=\sum_{i\geq 1}p_i<\infty$, and let $\pi_\lambda$ be the Poisson distribution…

Corrected Chernoff-bound conjecture. The function $B_L(z)$ is an upper bound for the lower-tail probability:

Modified DKWM conjecture. For any $m\ne n$,

Finite-sampling empirical-process conjecture. There exist constants $C,D>0$ (possibly $C=1$ and $D=2$?) such that the corresponding finite-sampling empirical-process tail bound hol…

Gaussian correlation conjecture. For all convex and centrally symmetric sets $C_1,C_2\subseteq\mathbb{R}^n$,

Extension conjecture. The displayed inequality holds throughout $C'$, even though $g_{p_1,p_2}$ is not unimodal for $\langle p_1,p_2\rangle\in C'\setminus C$. The conjecture is bas…