18 problems

Let a staircase tableau have size $n$, and let the $k$th diagonal be the diagonal whose parameters are being counted. The expected number of parameters on each diagonal is known. A…

Normal-limit conjecture. The distribution of the $r$-loadings of $\mathbf t$ converges to a normal distribution as $n\to\infty$.

Let $p=p_n\to\infty$ and suppose that the samples are generated from the ultra high-dimensional Gaussian populations and covariance structures considered in the paper, including th…

Let $T(x_1,\ldots,x_n):=2[h_1(x_1,\ldots,x_n)-h_0(x_1,\ldots,x_n)]$ be the nonnegative likelihood-ratio test statistic, where $h_0$ and $h_1$ are the hypotheses' maximized log-like…

Endo's asymptotic-distribution conjecture. For arbitrary $\Omega$, the probability of configuration $x$ in the asymptotic state is given by

Universality conjecture. The higher-order random limit is universal for all vertex-weight distributions with bounded support.

Normality conjecture. The numbers of perfect matchings, Hamilton cycles, spanning trees, and $k$-factors where $k\le d-1$, are all asymptotically normally distributed in…

Let $N$ be the sample length, let $\Theta_0$ be the null parameter subspace, and consider the single-regression Granger-causality estimator when the true non-null parameter approac…

Let $\IGamma_{yy}$ and $\bGamma_{yy|x}$ be the matrices arising from the single-regression Granger-causality estimator, and let $\lambda_i$ denote the eigenvalues of…

Gaussian-limit conjecture. The same result should hold for any concave $F$; that is, generally

Non-normality conjecture. In this scenario, the sampling distribution of $\sqrt{n}\{\tau(g,\hat f_n)-\tau(g,f)\}$ does not converge to a normal limit with mean $0$ and the efficien…

Let $T_n$ be a plane-oriented recursive tree with $n$ vertices, and let its Zagreb index be the sum of the squares of the degrees of its vertices. Non-Gaussian Zagreb-index conject…

Fixed-$d$ normality conjecture. For fixed $d$, the distribution of $X_{d,n}$ is asymptotically normal. That is,

Two-scale normality conjecture. For each fixed $n$, the distribution of $X_{d,n}$ is not asymptotically normal; in fact, $(X_{d,n}-\mu_{d,n})/\sigma_{d,n}$ tends to some abnormal d…

Asymptotic normality conjecture. The distribution is asymptotically normal as $n$ approaches infinity.

Asymptotic Gaussianity conjecture. The pair $(\nu(\bar{\tau}),\mu(\bar{\tau}))$ is also asymptotically Gaussian whenever

High-dimensional alternative-distribution conjecture. In the high-dimensional case, there could be new results regarding the limiting distributions of likelihood ratio test statist…

Asymptotic distribution conjecture. Almost surely, $G$ has a 3-star factor, and