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 $\widehat R$ 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\...
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 abso...
We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+\Sigma^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $\Sigma$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered entries with variance $N^{-1...
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{\lamb...
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 fa...
We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=\sigma_t \xi_t \in \mathbb{R}^N,$ where the coordinates of $\xi_t$ are i.i.d.\ and the scalar mixture variable $\sigma_t$ is shared by all coordinates. Under na...
J. Bouchaud, Pierre Bousseyroux, Tomas Espana et al.· 0 citations
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