For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \to \lambda \in (0,\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \exp(\sqrt{\lambda v}\, Z - \lambda v/2)$, where $v = \mathbb{E} X^4 - 1$ and $Z$ is standard normal. If $\binom{n}{d}/N \to c \in (0,\infty)$, the sample covariance of $N$ independent copies of $x$ has an almost-sure limiting spectral law: the free compound-Poisson law with rate $1/c$ and jump distribution $\operatorname{Law}(cR)$. It reduces to Marchenko-Pastur when $\lambda v = 0$. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when $\mathbb{E} X^3 \neq 0$; length and direction need not be independent.
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under symmetry and regular variation assumption with index $\alpha$ and any growth rate of the dimension, we prove that the universal CLT remains valid for $\alpha>3$. For $\alpha<3$, we identify a critical dimension growth at which the fluctuations of $\operatorname{tr}({\mathbf R}^2)$ become non-Gaussian. Moreover, if the dimension $p$ grows faster and $\alpha\le 3$ we establish a non-universal CLT with norming sequences depending on the value of $\alpha$. Our findings are illustrated in a simulation study.
Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ with a common covariance matrix given by the Kronecker product of $k$ unknown positive-definite factors, and let $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. Franks et al. (2026) established condition-number-free guarantees for the tensor-normal maximum likelihood estimator under the sample-size condition $nD\gtrsim k^2 d_{\max}^3$ and asked whether the cubic dependence on $d_{\max}$ could be reduced to a quadratic one. We answer this question affirmatively. For $t\geq 1$, if $nD\geq C k^2 d_{\max}^2 t^2$, then with high probability the maximum likelihood estimator exists, is unique, and satisfies $d_{\rm FR}(\widehat\Theta,\Theta)\leq C t \sqrt{k} d_{\max}/\sqrt{n}$ and $d_{\rm FR}(\widehat\Theta_a,\Theta_a)\leq C t\sqrt{k d_a} d_{\max}/\sqrt{nD}$ for every mode $a$. For every mode $a$ with $d_a=d_{\max}$, we further establish the sharp Thompson-metric bound $d_{\rm op}(\widehat\Theta_a,\Theta_a)\leq C t d_{\max}/\sqrt{nD}$. These guarantees are uniform over the unknown covariance factors and require neither condition-number bounds nor sparsity assumptions. Gaussian submodel lower bounds match the full and largest-factor Fisher--Rao rates up to a factor of $\sqrt{k}$ and the largest-factor Thompson rate up to universal constants. Consequently, for fixed $k$, the quadratic dependence of the sample-size threshold on $d_{\max}$ is optimal. GPT-5.6 Sol and Claude Fable 5 were used to assist with proof development, verification, and manuscript preparation.
Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.
Exhaustive moment fitting in this constant-dimensional space produces a proper mixture and, together with the dimension-free moment characterization of Gaussian mixtures, achieves the optimal Hellinger rate in polynomial arithmetic time for every fixed $k$.
The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular: its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. More generally, distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(\eta)$ law. For the uniform law, this yields a Chen-Stein proof of the extreme-correlation point-process limit and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. We also identify two distinct scales: $\eta_n\asymp n$ alters the limiting spectrum, whereas $\eta_n\asymp n^2$ is needed to keep the Frobenius distance bounded. Finally, for a bounded, centered i.i.d. off-diagonal specification, projection to the nearest correlation matrix incurs a squared repair cost asymptotically at least one-half of the squared Frobenius norm of its off-diagonal part.
We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $\omega$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak--\L ojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.
Matthew Rosenzweig, D. Slepčev, Lihan Wang· 1 citation
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