LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Independent lower bound for the largest sample-covariance eigenvalue

Statement

Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid array of real random variables with mean zero, variance one and finite fourth moment. For each positive integer $n$ , let $p_{n}$ be a positive integer and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq p_{n}}$ and suppose $p_{n} / n \to γ \in (0, \infty)$ . For $S_{n} = X_{n}^{T} X_{n} / n$ and $b_{γ} = (1 + \sqrt{γ})^{2}$ , the separate lower-edge estimate is ${lim inf}_{n \to \infty} λ_{\max} (S_{n}) \geq b_{γ} almost surely .$

Statement references

Yin, Bai, and Krishnaiah, Theorem 3.1, printed p. 517 (complete paper PDF).

Baik and Silverstein, Theorem 1.1 for the identity population case, printed pp. 4–5 (complete paper PDF).

Facts & Assumptions

[F1]

Nonbacktracking covariance high moments and an independent coarse norm bound supplies an almost-sure eventual finite bound on $∥ X_{n} / \sqrt{n} ∥$ , without using any sharp edge or empirical spectral limit.

[F2]

Fixed-degree covariance trace moments and independent edge witnesses proves the independent fixed-degree trace moments and upper-endpoint polynomial witness under the coarse bound in F1.

[F3]

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition identifies the positive spectra of $X_{n} X_{n}^{T} / n$ and $S_{n} = X_{n}^{T} X_{n} / n$ ; their largest eigenvalues are equal.

Proof

Given: The nested iid array and hypotheses in the Statement.

1.1givenF1F2

Apply F1 to the upper-left blocks. On its probability-one event the hypothesis of F2 is satisfied. F2 therefore gives ${lim inf}_{n} λ_{\max} (X_{n} X_{n}^{T} / n) \geq (1 + \sqrt{γ})^{2}$ almost surely. Its argument uses fixed polynomial trace witnesses derived from tree counting and bounded truncations; it invokes no weak ESD convergence or limiting-support theorem.

2.1F3step 1.1∎

Transfer the largest eigenvalue through the SVD identity in F3. This yields ${lim inf}_{n} λ_{\max} (S_{n}) \geq b_{γ}$ almost surely, as asserted.

Depends on

Used by

Almost-sure limit of the largest sample-covariance eigenvalue Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yin, Bai, and Krishnaiah, On the Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix, Theorem 3.1 (standard reference, not scraped)
Baik and Silverstein, Eigenvalues of Large Sample Covariance Matrices of Spiked Population Models, Theorem 1.1 (standard reference, not scraped)