26 problems

Fix $m\geqslant2$ and $\delta>0$. Let $T_n$ be the number of triangles in the instantaneous variant of the linear preferential attachment model, with connections specified by the i…

Janson's conjecture. A combination of Stein's method for normal approximation and the Chen--Stein method for Poisson approximation might be used to prove the same asymptotic normal…

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with mean $0$ and variance $\sigma^2$, let $S_n=X_1+\cdots+X_n$, and write $P_{S_n}$ for the law of…

Triple-sum normal approximation conjecture. Under these modified locality and stabilization definitions, it should be possible to control the difference operators associated with t…

The Kolmogorov-distance rate conjecture. The same convergence rate as for the distance $d_1$ should hold for the Kolmogorov distance $d_K$, namely

Let $m\geqslant 2$ be fixed, and let $\Lambda$ be a fixed tree. Write $W_n$ for the number of copies of $\Lambda$ in the uniform attachment graph $G^m_n$. Normality conjecture for…

Rio's conjecture. The optimal bound

Let $S_{n,k}$ and $S'_{n,k}$ be the binomial and Poisson $k$-cluster counts, respectively, and let $b:=\limsup_{n\to\infty}(n\theta r^d/\log n)$. For the normal approximations in t…

Let $Y_i$ have mean $\mu$ and standard deviation $\sigma$, and define its skewness and excess kurtosis by … assuming that these quantities exist and are finite. Let $e^*(n,z)$ deno…

Let $Y_i$ have distribution function $F_{Y_i}$, and let $e^*(n,z)$ denote the approximation error of the central limit theorem for the standardized sum, with sample size $n$ and st…

Consider the setting of Theorem Trate: a constrained or unconstrained U-statistic built from the random sequence and kernel in the paper, with Kolmogorov distance $d_K$ and the the…

Pemantle's conjecture. If $\sigma(X)\gg1$, then $X$ is approximately normally distributed. This conjecture was the original motivation for studying anti-concentration from zero-fre…

For a locally finite point set in the region … let the maximal layers be induced by a homogeneous Poisson point process, where $F:[0,\alpha^{1/(d-1)}]^{d-1}\to[0,\infty)$ has conti…

Optimal normal-approximation rate. The error rate

Let $X_1,\ldots,X_n$ be independent $p$-dimensional random vectors with independent and identically distributed components, and let $\mathcal{A}^{re}$ denote the class of $p$-dimen…

Let $p$ be a positive integer and let $W=\sum_{i\in I}X_i$, where ${\mathbbm{E}}X_i=0$, ${\mathbbm{E}}W^2=1$, and the variables satisfy (LD1)--(LD$(p+1)$): for each $i_1\in I$,…

Asymptotic exactness conjecture. In general, this lower bound is at least asymptotically exact, meaning that the lower bound captures the asymptotic behaviour of $\varepsilon_n$.

Improved Kolmogorov bound. There is an absolute constant $C$ such that

Let $X$ be a multiple Wiener--Itô integral of order $p\geq 2$, and let $\kappa_6(X)$ denote its sixth cumulant. Sixth-cumulant positivity conjecture. For every such $X$, one has ……

Let $k,l\geq 2$ be distinct positive integers, and let $\{X_n\}_{n\geq 1}$ be a sequence of eigenfunctions lying in the same eigenspace of a Markov generator $\mathrm{\mathbf L}$ s…

Let $M$ be a binomially or Poisson-distributed random variable with $G$-transform $G(M)$, and let $\Phi$ denote the standard Gaussian distribution function. The increasing property…

Let $M$ be a binomially distributed or Poisson-distributed random variable, let $P_0$ be its distribution, and let $G(M)$ be its $G$-transform, defined by assigning the signed squa…

The sharpened Tusnády inequality. The bounds

Logan et al.'s conjecture. The sequence $\psi_n(\mathbf{1},\mathbf{Y})$ is asymptotically normal if and only if $\mathbf{Y}$ is in the domain of attraction of the normal law.

Let $S_n$ be the normalized sum of independent Rademacher variables, and let $Z$ be a standard normal random variable. Efron's conjecture. There exists an upper bound for…