Let $\widehat R$ be the Pearson sample correlation matrix formed from $n$ independent Gaussian observations in $p$ dimensions, and write $m=n-1\ge p$. Under the null correlation $R=I_p$, the classical independent beta product, exact cumulants, and full Fourier inversion yield, along every sequence $p\to\infty$ with $m\ge p$, a uniform first Edgeworth expansion for $\log\det\widehat R$, centered by its exact mean and scaled by its exact standard deviation. The expansion identifies the exact finite dimensional skewness correction and gives the sharp Kolmogorov equivalent $A_{m,p}/\{6\sqrt{2\pi}V_{m,p}^{3/2}\}$, where $V_{m,p}$ is the exact variance and $A_{m,p}$ is the absolute third cumulant. This equivalent unifies the square, fixed gap, growing gap, proportional, and dilute regimes; in the square regime the error has order $(\log p)^{-3/2}$ with an exact constant. For every positive definite population correlation matrix $R$, we prove a uniform finite sample Berry-Esseen bound that explicitly tracks population dependence. All theoretical results have exact or proved equivalent Lean 4 formulations whose declarations and dependencies are kernel checked.
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.
We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$. Uniformly over deterministic compact intervals $I\subseteq I_n$ whose length $\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\log\ell/\sqrt\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\ell\to\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.
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.
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 study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost-sure convergence of $\lambda_n (\mathbf{R}_n)$ to the lower Mar\v{c}enko--Pastur edge $\lambda_- = (1 - \sqrt{\phi})^2$ without additional moment assumptions. In the wide regime $\phi<1$, we establish a phase transition at the third-order tail scale. If $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to 0$ as $t \to \infty$, the smallest eigenvalue $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to $\lambda_-$. While if $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to \infty$, then $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to zero. At the critical scale $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to \kappa \in (0, \infty)$, the point process of eigenvalues in the lower gap $(0, \lambda_-)$ converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of $\lambda_{p_n} (\mathbf{R}_n)$, which has a continuous density on $(0, \lambda_-)$ and a positive atom at $\lambda_-$.
Ze-Qin Lin, Guamgming Pan, Hao-Zhu Zhao et al.· 0 citations
Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$, and let $X$ be standard Gaussian. Because Gaussian observations are invariant under $O(d)$, the natural inverse problem is to recover the orthogonal orbit of $P$; the law of $P(X)$ alone is generally insufficient. We prove that a prescribed finite family of mixed moments of correlated Gaussian replicas, $$ M_{P,r}(\Sigma)=\mathbb{E}\prod_{a=1}^r P(X_a), $$ separates $O(d)$-orbits. We construct an explicit replica cutoff and rational covariance grids satisfying $$ \frac{1}{2}I_r\preceq\Sigma\preceq\frac{3}{2}I_r. $$ Finite differences recover all complete Wick contractions needed by invariant theory, giving an exact finite decoder. The resulting probe map is bi-H"older equivalent to orbit distance on coefficient balls, with an effective exponent. We then identify the same certificate in an irregular period system. Replicated characteristic functions are polynomial oscillatory periods, and their mixed derivatives at zero are the moments above. If the leading homogeneous part of $P$ has an isolated critical point, the active-replica face indexed by $I$ has twisted de Rham rank $(m-1)^{d|I|}$; zero coupling is therefore a rank-changing boundary. The forced scaling $$ \tau_a=\rho^{m-2}\lambda_a,\qquad x_a=\rho^{-1}u_a $$ produces compatible Rees--Jacobi lattices and, under central nonresonance, a canonical rank-one Gaussian branch. On admissible tame Morse chambers, the period matrix factors into algebraic Jacobi, sectorial thimble, and integral Betti components. Projecting the assembled real-contour period onto the Gaussian branch recovers exactly the finite orbit certificate.
O. J. Assaad· 0 citations
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