16 problems

Eaton's conjecture. For $x>\sqrt{2}$,

Let $V(t)$ denote the random variable measuring the quantity studied in the preceding estimates for the excited random walk at time $t$, and let $C,c>0$ be constants. Square-expone…

Exponential-moment conjecture. In the supercritical case $alpha>1$, the random variable $(R^*)^d$ has a finite exponential moment; equivalently, there exists $c>0$ such that

Jordan lower-tail transition conjecture. The lower tail of the Jordan-$2$ ordering undergoes a transition from polynomial decay to stretched-exponential decay, and a similar phenom…

Let $m$ be the sampling number, let $\lambda=\lVert A\rVert_2$ and $\phi=\lVert A\rVert_F$ for a symmetric matrix $A$, and let $\varepsilon_{\mathrm{abs}}$ be the threshold appeari…

Let $m$ be the sampling number, let $mu=r_{\mathrm{eff}}(A)$ for a nonzero symmetric positive semidefinite matrix $A$, and let $\varepsilon_{\mathrm{rel}}$ be the threshold appeari…

Fix $lambda>0$ and $phi>0$, and let $\hat{x}_{\mathrm{upper}}$ be the supremum of the inflection points defined in Theorem 2 for the absolute-error Gamma family. Absolute-tail boun…

Let $\bar{x}_{\mathrm{upper}}$ and $\bar{x}_{\mathrm{lower}}$ be the points defined in Theorem 1 for the relative-error Gamma family, with effective shape parameter $\mu$. Extremal…

Let $B_{n,k}(p)$ be the binomial tail probability, $b_{n,k}(p)$ the corresponding binomial point probability, and $L(n,k,p)$ the lower bound from the paper's Theorem. Suppose…

Conjectured sharp local tail bound. The same inequality should hold with the power $-1$ in place of $\frac{d}{2}-1$:

Oleszkiewicz's conjecture. There is a universal constant $c$ such that, for every real $u$,

Let $p$ denote the parameter in the binomial distribution, and let Inequality … can be significantly improved when $p=\frac{1}{2}$. This conjecture motivated the research in the pa…

Let $\{-1,1\}^n$ carry the uniform probability measure, and let the random-walk semigroup on the discrete cube play the role of the Ornstein–Uhlenbeck semigroup. For a non-negative…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, and let $(Q_t)$ be the Ornstein–Uhlenbeck semigroup defined by … For a non-negative function $f$ of integral $1$,…

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, with $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$, and let … where…

Let $X_i=a_i\eta_i$ for independent, not necessarily identically distributed, zero-mean random variables $\eta_i$ satisfying $|\eta_i|\leqslant1$ almost surely, with…