14 problems

Real sample covariance conjecture. For real sample covariance, Theorem 1 should still hold, with different limiting distributions but the same scaling. In particular, the critical…

Let $\{w_j\}_{j=1}^{p}$ be a deterministic sequence of real values such that $|w_j|\le 1$ for all $j\in[\![p]\!]$. Let $\{\mathbf{v}_j\}_{j=1}^{p}$ be the eigenvectors of $\Sigma$,…

Let $\widehat\Sigma_N=\frac{1}{N}\sum_{k=1}^N\mathbf{x}_{pk}\mathbf{x}_{pk}^{\top}$ be the sample covariance matrix of independent copies of a random vector following the random te…

Rate-function regularity conjecture. $I(x)$ is smooth at every point $x\neq x_c(\rho)$, and when $\eta\in[1/k,1/(k-1))$, the first discontinuous derivative of $I$ at $x=x_c(\rho)$…

Let $M<N$, let $\mathbf u\in\mathbb C^M$ be a fixed unit vector, and define … Let $F_{1c}$ be the asymptotic empirical spectral distribution of $Q_1$, let $F_{1c}^{-1}$ be its quan…

Let $Q_1$ be a sample covariance matrix with non-scalar population covariance $\Sigma$, and let $\xi_k$ denote its eigenvectors. For each $k$, let $\mathbf D_k$ be a covariance mat…

Let $\xi$ be a discrete real random variable satisfying a non-degeneracy condition, such as the condition denoted by … should extend to this discrete setting: with probability $1$,…

Optimal-rate conjecture. The optimal convergence rates should be $O(N^{-1/2})$ for $H^{\mathbf{S}_n}(x)$ and $O(N^{-1})$ for $F^{\mathbf{S}_n}(x)$.

Asymptotic Haar eigenmatrix conjecture. If $B_{s,t}^{n}$ weakly converges to a Brownian bridge, this provides evidence that the eigenmatrix $U_n$ is asymptotically Haar distributed…

Let ${\bf A}_n=\frac{1}{n}{\bf X}_n{\bf X}_n^T$, where ${\bf X}_n=(X_{ij})_{p\times n}$ has independent and identically distributed real entries with $\mathbb{E}X_{11}=0$ and…

Asymptotic independence conjecture. The fluctuations of $\hat{\lambda}_1$ and $\hat{\lambda}_N$ under $\mathcal H_0$ are asymptotically independent; consequently, the false-alarm c…

Chafaï's conjecture. The sequence $\mathcal W^{(n)}$ converges in distribution to the tied-down Brownian bridge, with the same normalization as in Theorem 1:…

Asymptotic Haar eigenvector conjecture. The matrix of sample eigenvectors should be asymptotically Haar distributed.

Let $Q$ be a distribution on the real line with expectation $0$, variance $1$, and finite fourth moment, and let $X$ be a sample matrix with independent identically distributed ent…