CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Fixed negative-axis directional resolvent integrals converge

Statement

Under the hypotheses of the sample-covariance resolvent deterministic-equivalent theorem, for each fixed $s > 0$ the directional measures satisfy $\int_{[0, \infty)} \frac{1}{t + s} d μ_{n}^{x_{n}} (t) ⟶ \int_{[0, \infty)} \frac{1}{t + s} d F_{c, H} (t) almost surely .$

Statement references

Bai, Miao, and Pan, Theorem 1, printed pp. 8–9 (full paper PDF).

Facts & Assumptions

Given: The hypotheses and directional measures in the Statement.

[F1]

The directional measures converge almost surely weakly to $F_{c, H}$ (Sample-covariance resolvent deterministic equivalents).

[F2]

Weak convergence means convergence of integrals against every bounded continuous real-valued function (Weak convergence of borel probability measures).

[F3]

The directional spectral measure is $\sum_{λ} ∥ P_{λ} x_{n} ∥_{2}^{2} δ_{λ}$ for a deterministic unit vector (Directional spectral measure of a Hermitian matrix).

Proof

1.1givenF3algebra

Fix $s > 0$ . The function $f_{s} (t) = (t + s)^{- 1}$ is continuous on $[0, \infty)$ and satisfies $0 < f_{s} (t) \leq 1 / s$ , so it is bounded. The measure is the directional spectral law in [F3].

2.1givenF1F2step 1.1∎

On the probability-one event in [F1], apply the defining bounded-continuous-test property [F2] to $f_{s}$ to obtain the asserted integral convergence.

Source notes

The proof is the direct bounded-continuous-test consequence of the preceding directional weak-convergence theorem.

Depends on

Used by

At aspect one the reciprocal-spectrum limit diverges Example

Dependency tree · two levels

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Sources

Z. D. Bai, B. Q. Miao, and G. M. Pan, On Asymptotics of Eigenvectors of Large Sample Covariance Matrix (standard reference, not scraped)