LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

A coarse complex sample-covariance norm bound from real edge bounds

Statement

Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid array of centered complex random variables with $E ∣ ξ_{11} ∣^{2} = τ^{2} < \infty$ and $E ∣ ξ_{11} ∣^{4} < \infty$ . For positive integers $n$ , let $N_{n}$ be positive with $n / N_{n} \to c \in (0, \infty)$ , and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq N_{n}}$ . Then almost surely ${lim sup}_{n \to \infty} \frac{∥ X_{n} ∥_{o p}}{\sqrt{N_{n}}} \leq (σ_{R} + σ_{I}) (1 + \sqrt{c}), σ_{R}^{2} = E (Re ξ_{11})^{2}, σ_{I}^{2} = E (Im ξ_{11})^{2} .$ In particular, when $τ^{2} = 1$ , the right side is at most $\sqrt{2} (1 + \sqrt{c})$ .

Statement references

Yin, Bai, and Krishnaiah, Theorem 3.1, as reconstructed in the current real finite-fourth-moment upper-edge supplier (full paper PDF).

Bai, Miao, and Pan, Section 2, printed pp. 12–16 (full paper PDF).

Facts & Assumptions

Given: The common nested complex iid array and its centered finite-fourth-moment entry law.

[F1]

Iid entries are independent and share one common distribution (Identical distribution and IID families).

[F2]

For a real nested iid array with mean zero, variance one and finite fourth moment, $lim sup λ_{\max} (Z_{n}^{T} Z_{n} / n) \leq (1 + \sqrt{γ})^{2}$ almost surely when the number of columns divided by the number of rows tends to $γ > 0$ (Finite-fourth-moment upper bound for the largest sample-covariance eigenvalue).

[F3]

The operator norm is the largest singular value, and the largest eigenvalue of $Z^{*} Z$ is its square (The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector, Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F4]

The operator norm is the maximum of $∥ A v ∥_{2}$ over unit vectors in the domain (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

Proof

1.1givenF1F2F3algebra

Write $κ_{n} = N_{n} / n \to 1 / c$ , $σ_{R}^{2} = E (Re ξ_{11})^{2}$ , and $σ_{I}^{2} = E (Im ξ_{11})^{2}$ . Centering of $ξ_{11}$ makes both real coordinates centered, and $σ_{R}^{2} + σ_{I}^{2} = τ^{2}$ . If $σ_{R} > 0$ , the matrices $Z_{n} = Re X_{n} / σ_{R}$ are upper-left blocks of one real iid array with mean zero, variance one and finite fourth moment. Applying [F2] with $γ = κ = 1 / c$ and then [F3] gives ${lim sup}_{n} \frac{∥ Re X_{n} ∥_{o p}}{\sqrt{N_{n}}} \leq σ_{R} \frac{1 + \sqrt{κ}}{\sqrt{κ}} = σ_{R} (1 + \sqrt{c}) almost surely .$ If $σ_{R} = 0$ , the centered real part is zero almost surely, so the same bound holds. The identical argument for the imaginary part gives the bound with $σ_{I}$ .

2.1givenF4step 1.1algebra∎

The complex operator norm satisfies $∥ X_{n} ∥_{o p} = ∥ Re X_{n} + i Im X_{n} ∥_{o p} \leq ∥ Re X_{n} ∥_{o p} + ∥ Im X_{n} ∥_{o p}$ . Intersecting the two probability-one events from step 1.1 proves the asserted limsup bound. Finally, $σ_{R} + σ_{I} \leq \sqrt{2 (σ_{R}^{2} + σ_{I}^{2})} = \sqrt{2} τ$ by Cauchy–Schwarz, giving the unit-variance consequence.

Source notes

The real edge application uses the current local upper-edge proof for the same top-left nested blocks, finite-fourth-moment law, dimension ratio and normalization. The complex-to-real reduction and triangle bound are proved above.

Depends on

Used by

Sample-covariance resolvent deterministic equivalents Theorem

Dependency tree · two levels

16 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, Theorem 3.1 (real upper-edge result) (standard reference, not scraped)
Z. D. Bai, B. Q. Miao, and G. M. Pan, On Asymptotics of Eigenvectors of Large Sample Covariance Matrix (standard reference, not scraped)