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Multivariate analysis

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let the $n$ columns $M_j\in\mathbb R^p$ be independent with law $N_p(\mu,\Sigma)$, where $\Sigma$ is positive definite with eigenvalues $\eta_1\ge\cdots\ge\eta_k>1$ and $\eta_{k+1}=\cdots=\eta_p=1$, with $k$ fixed as $p,n\to\infty$. Put $\bar M=n^{-1}\sum_j M_j$, $\widehat M_j=M_j-\bar M$, $S=n^{-1}\widehat M\widehat M^T$, and write $l_1\ge\cdots\ge l_p\ge0$ for the sample eigenvalues. Assume $p/n\to\gamma\in(0,\infty)$.

Statement recorded in the supplied source.
Determine the asymptotic distributions of the leading sample eigenvalues $l_1,l_2,\ldots$ for these real spiked populations, identifying their appropriate centerings, scalings and joint laws throughout the different spike regimes, including the transition at $\eta=1+\sqrt\gamma$. [S2]

Short English statement

Find the limiting eigenvalue statistics for high-dimensional real Gaussian covariance matrices with finitely many population spikes, including phase transitions.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-88-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-88

Multivariate analysis

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let the $n$ columns $M_j\in\mathbb R^p$ be independent with law $N_p(\mu,\Sigma)$, where $\Sigma$ is positive definite with eigenvalues $\eta_1\ge\cdots\ge\eta_k>1$ and $\eta_{k+1}=\cdots=\eta_p=1$, with $k$ fixed as $p,n\to\infty$. Put $\bar M=n^{-1}\sum_j M_j$, $\widehat M_j=M_j-\bar M$, $S=n^{-1}\widehat M\widehat M^T$, and write $l_1\ge\cdots\ge l_p\ge0$ for the sample eigenvalues. Assume $p/n\to\gamma\in(0,\infty)$.

Statement recorded in the supplied source.
Determine the asymptotic distributions of the leading sample eigenvalues $l_1,l_2,\ldots$ for these real spiked populations, identifying their appropriate centerings, scalings and joint laws throughout the different spike regimes, including the transition at $\eta=1+\sqrt\gamma$. [S2]

Short English statement

Find the limiting eigenvalue statistics for high-dimensional real Gaussian covariance matrices with finitely many population spikes, including phase transitions.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Exact reduction of mean-centered real spiked covariance to an uncentered Wishart matrix with n-1 samples; joint spike-regime laws remain unproved.
Sources and literature check.
Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.
Removing the empirical mean exactly.
This attempt checks a normalization issue before any spike asymptotics. Let $Z$ be a $p\times n$ matrix of independent standard normal entries, and write $M=\mu\mathbf1^T+\Sigma^{1/2}Z$. With $P=I_n-n^{-1}\mathbf1\mathbf1^T$, the centered matrix is $\widehat M=\Sigma^{1/2}ZP$. Choose an orthogonal matrix whose last column is $n^{-1/2}\mathbf1$. Orthogonal invariance of each Gaussian row shows that $ZO$ again has independent standard normal entries. Consequently \[ S\ \stackrel{d}=\ \frac1n\Sigma^{1/2}XX^T\Sigma^{1/2} =\frac{n-1}{n}\widetilde S, \qquad \widetilde S=\frac1{n-1}\Sigma^{1/2}XX^T\Sigma^{1/2},\] where $X$ is $p\times(n-1)$. This identity is exact, including when $p\ge n$; in that case the rank is at most $n-1$ and there are the corresponding zero sample eigenvalues.
What this transfers and what it does not.
Suppose a theorem for the uncentered model supplies a deterministic centering $c_{p,n-1}$, positive scale $a_{p,n-1}$, and a joint limit for a finite list of eigenvalues. Multiplying both center and scale by $(n-1)/n$ transfers that theorem exactly to the stated centered convention. One must use aspect ratio $p/(n-1)$ in the uncentered theorem. For example, an $O(n^{-1})$ centering correction is negligible at scale $n^{-2/3}$ or $n^{-1/2}$, provided its coefficient is uniformly bounded; this follows from the respective ratios $n^{-1/3}$ and $n^{-1/2}$ tending to zero. It need not be discarded before checking a critical spike tuning.
First gap.
The calculation removes neither the spectral interaction among spikes nor the critical transition. It does not derive any joint edge or outlier law, and it assumes no theorem about those laws. Deriving the required limiting distribution for $\widetilde S$, uniformly in the relevant critical tuning, remains the first missing step. The elementary Gaussian reduction is known in substance and is not counted as new progress on the complete distributional problem.
Verification and adversarial audit.
Checked Gaussian orthogonal invariance, denominator n rather than n-1, p>=n zero eigenvalues, and scale comparisons; did not import unverified spike limit theorems. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked Gaussian orthogonal invariance, denominator n rather than n-1, p>=n zero eigenvalues, and scale comparisons; did not import unverified spike limit theorems. Scope remains limited: Obtain the joint eigenvalue asymptotics of the reduced uncentered ensemble, including critical spike tuning.

Coordinator checked the Gaussian row rotation, rank n-1 and exact centered-Wishart prefactor. The critical spike law is not supplied by this reduction. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Obtain the joint eigenvalue asymptotics of the reduced uncentered ensemble, including critical spike tuning.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

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