28 problems

A partition is an $s$-core partition if none of its hook lengths is divisible by $s$; an $(s_1,s_2,\cdots,s_m)$-core partition is simultaneously an $s_i$-core partition for every…

Minimal sparsity conjecture. The condition

Asymptotic normality conjecture. In this case, $X_n$ is typically asymptotically normal, possibly with a $d$-periodic variance. This is conjectured based on the fact that asymptoti…

Let $H$ be a graph with a self-gluing $d711: U_1 \to U_2$, and let $K$ be a valid minimal cut for this self-gluing. Construct the swapping $(H',\psi)=\operatorname{Sw}(H,d711,K)$ b…

Let $T$ be the corrected least-squares test statistic … Here $\hat{\boldsymbol{\beta}}^{\textrm{CLS}}$ is the corrected least-squares estimator, $\boldsymbol{e}_j$ is the $j$th sta…

Sample PLS asymptotic normality conjecture. The sample PLS is asymptotically normal with

Let $U_{n,m,q}$ be the full two-sample U-statistic and let $U_{n,m,q}^M$ be its monotonically indexed counterpart, defined from two independent samples $X_1,\ldots,X_n$ and…

Let $U_{n,q}^{M,(qs)}$ denote the $(qs)$-th Hoeffding component of the monotonically indexed statistic $U_{n,q}^M$, and let $\boldsymbol\Sigma_s(q)$ be the corresponding covariance…

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

Let $a_* = \bm{\hat\beta}^T\bm\Sigma\bm w$ be the random correlation parameter, and let $\bar a$ be its deterministic limit in probability obtained by solving the nonlinear system…

Let $\boldsymbol{\tau}_n$ be a uniformly random tree permutation of length $n$, and let $\boldsymbol{\tau}_n$ have a fixed forest pattern $\sigma$ with $d\geqslant2$ blocks, each o…

Let $Y$ and $\mZ$ be independent, and let $\xi_n^\#$ be the coefficient defined by … Here $R_i$ and $M(i)$ are the rank and nearest-neighbor index quantities defined for the covari…

Let $\widehat{{{v}}}_2$ and $\widehat{{{u}}}_2$ be the estimated eigenvectors, and let $v_{i2}$ and $u_{i2}$ denote their corresponding population eigenvector entries for each fixe…

Let $\xi$ be an offspring distribution with $\operatorname{\mathbb E}\xi=1$, and let $\mathcal T_n$ be a conditioned Galton–Watson tree with offspring distribution $\xi$. For any f…

Let an ARCH process be a time-series process with autoregressive conditional heteroskedasticity, and let the marginal law of its observations have a finite fourth moment. Let Theor…

General skew-shape asymptotic normality conjecture. Asymptotic normality for general skew shapes, rather than only block diagonal skew shapes, holds if and only if

Let $k,n$ tend to infinity. Consider the estimators $\widehat{\mathbb{IF}}_{22,k}([\widehat{\Omega}_{k}^{\text{shrink}}]^{-1})$ and…

Let $R_{n,k}$ count records through time $n$ that break precisely $k$ remaining records, and write … For fixed $k\geq0$, sharper variance asymptotics conjecture. There exist consta…

Let $\Psi$ be the graph functional summing a component functional $\psi$ over all components, and let $\Psi^\dagger$ denote the corresponding functional with the contribution from…

Let $G(n,\mathbf d_n)$ and $G^*(n,\mathbf d_n)$ be the random graphs with the given degree sequence considered in the paper, and let $\mathcal C_1$ denote their giant component. Th…

Asymptotic normality conjecture.

For each $n\geq 0$, let $G_n$ be an acyclic graph on $n$ vertices, and write ${G_n \brace k}$ for the number of partitions of the vertex set of $G_n$ into $k$ non-empty independent…

Let $\widehat{\tau_{HT}}(d_k,d_l)$ be the Horvitz–Thompson estimator of the average causal effect for exposure conditions $d_k$ and $d_l$. The asymptotic growth process is allowed…

Let $(F^{c(n)}_n)_{n geq 0}$ be a sequence of forests, where $F^{c(n)}_n$ has $n$ vertices and $c(n)$ components. Let $X^{c(n)}_n$ be the number of classes in a uniformly chosen pa…

Let $\widehat{\mathbf{B}}_n$ be the MM estimate of the coefficient matrix in the multivariate linear model, based on a random sample of size $n$, and let (A4)–(A7) denote the assum…