LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Adapted concentration of bounded complex resolvent quadratic forms

Statement

On a common probability space, for each positive integer $n$ , let $X_{n} \in C^{n \times N_{n}}$ have entries from one common iid law with mean zero, unit second absolute moment, and absolute value at most $K$ , where $K$ is independent of $n$ . Let $T_{n} \in C^{n \times n}$ be deterministic Hermitian positive semidefinite with $∥ T_{n} ∥_{o p} \leq M$ , and let $x_{n} \in C^{n}$ be deterministic with $∥ x_{n} ∥_{2} = 1$ . Assume $N_{n} / n \to κ \in (0, \infty)$ . For fixed $z = u + i η$ with $η > 0$ , set $A_{n} : = \frac{1}{N_{n}} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}, R_{n} (z) : = (A_{n} - z I)^{- 1} .$ Then $x_{n}^{*} R_{n} (z) x_{n} - E [x_{n}^{*} R_{n} (z) x_{n}] \to 0$ almost surely. Moreover, for every $t > 0$ there is a constant $C = C (K, M, z, κ)$ such that, for all sufficiently large $n$ , $P ⁣ {∣ x_{n}^{*} R_{n} (z) x_{n} - E [x_{n}^{*} R_{n} (z) x_{n}] ∣ > t} \leq \frac{C}{t^{4} N_{n}^{2}} .$

Statement references

Bai, Miao, and Pan, §2, equations (2.1)–(2.3), printed pp. 12–13 (full paper PDF).

Silverstein, Theorem 1.1 proof, PDF pp. 3–8 (full paper PDF).

Facts & Assumptions

Given: The common probability space, bounded iid entries, deterministic PSD $T_{n}$ , unit vectors $x_{n}$ , and fixed $z \in C^{+}$ in the Statement.

[F1]

The entries are mutually independent with the stated common law (Identical distribution and IID families).

[F2]

Independent columns have their product joint law (Independent random elements have product joint law).

[F3]

Fubini permits the finite product-integral conditional averages used for the column filtration (Fubini's theorem for L^1 functions on a sigma-finite product).

[F4]

The Hermitian rank-one resolvent update and nonzero denominator are given by Hermitian rank-one resolvent update and trace-norm bound.

[F5]

A summable sequence of probabilities has probability zero of occurring infinitely often (First Borel-Cantelli lemma for events).

[F6]

A finite-dimensional Hermitian PSD matrix has nonnegative real eigenvalues and a unitary eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F7]

For a nonnegative random variable $Y$ and $a > 0$ , $P (Y > a) \leq E [Y] / a$ (Markov's inequality for random variables).

[F8]

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

Given: The hypotheses and notation in the Statement.

1.1givenF1algebra

We first prove a linear-form bound. If $ζ_{i}$ are independent centered complex variables with $∣ ζ_{i} ∣ \leq K$ and $a_{i} \in C$ , expansion of $E ∣ \sum_{i} a_{i} ζ_{i} ∣^{8}$ has zero contribution whenever an index occurs exactly once. Each remaining equality pattern partitions the eight positions into blocks of sizes at least two. For a fixed pattern, the absolute sum is at most $K^{8} \prod_{B} \sum_{i} ∣ a_{i} ∣^{∣ B ∣} \leq K^{8} (\sum_{i} ∣ a_{i} ∣^{2})^{4}$ , since $\sum_{i} ∣ a_{i} ∣^{r} \leq (\sum_{i} ∣ a_{i} ∣^{2})^{r / 2}$ for $r \geq 2$ . There are finitely many such patterns, so $E ∣ \sum_{i} a_{i} ζ_{i} ∣^{8} \leq C_{K} (\sum_{i} ∣ a_{i} ∣^{2})^{4}$ .

2.1givenF2F4F6F8step 1.1algebra

Write $s_{n, j} = T_{n}^{1 / 2} X_{n, \cdot j} / \sqrt{N_{n}}$ , $A_{n, j} = A_{n} - s_{n, j} s_{n, j}^{*}$ , and $R_{n, j} = (A_{n, j} - z I)^{- 1}$ . By [F4], $U_{n, j} : = x_{n}^{*} (R_{n} - R_{n, j}) x_{n} = - \frac{(x_{n}^{*} R_{n, j} s_{n, j}) (s_{n, j}^{*} R_{n, j} x_{n})}{1 + s_{n, j}^{*} R_{n, j} s_{n, j}} .$ Diagonalizing $A_{n, j}$ as in [F6] gives $Im (z (1 + s_{n, j}^{*} R_{n, j} s_{n, j})) \geq η$ , hence the denominator has modulus at least $η / ∣ z ∣$ . Conditional on all columns except $j$ , each factor in the numerator is a linear form in the bounded independent entries of column $j$ , with coefficient-vector norm at most $\sqrt{M} / (η \sqrt{N_{n}})$ . Step 1.1 and Cauchy–Schwarz therefore give $E ∣ U_{n, j} ∣^{4} \leq C N_{n}^{- 4}$ .

3.1givenF2F3step 2.1algebra

Let $F_{n, j} = σ (X_{n, \cdot 1}, \dots, X_{n, \cdot j})$ , and let $E_{n, j}$ be the finite-product average over the remaining columns. By [F2]–[F3], $x_{n}^{*} R_{n} x_{n} - E [x_{n}^{*} R_{n} x_{n}] = \sum_{j = 1}^{N_{n}} (E_{n, j} - E_{n, j - 1}) x_{n}^{*} R_{n} x_{n} .$ The leave-one-column resolvent $R_{n, j}$ depends on all columns except $j$ , so its product-integral averages under $E_{n, j}$ and $E_{n, j - 1}$ agree. Therefore the $j$ th summand is $D_{n, j} : = (E_{n, j} - E_{n, j - 1}) U_{n, j}$ . Integrating over column $j$ shows $E_{n, j - 1} D_{n, j} = 0$ , so these are martingale differences. Conditional Jensen gives $E ∣ D_{n, j} ∣^{4} \leq 16 E ∣ U_{n, j} ∣^{4} \leq C N_{n}^{- 4}$ .

4.1step 3.1algebra

Put $S_{k} = \sum_{j \leq k} D_{n, j}$ , $a_{k} = E ∣ S_{k} ∣^{4}$ , $v_{k} : = E_{n, k - 1} ∣ D_{n, k} ∣^{2}$ , and $w_{k} : = E_{n, k - 1} ∣ D_{n, k} ∣^{4}$ . Expanding $∣ S_{k - 1} + D_{n, k} ∣^{4}$ , the term linear in $D_{n, k}$ vanishes after conditional integration; $∣ Re ({\overline{S}}_{k - 1} D_{n, k}) ∣ \leq ∣ S_{k - 1} ∣ ∣ D_{n, k} ∣$ and $4 ∣ S_{k - 1} ∣ ∣ D_{n, k} ∣^{3} \leq 2 ∣ S_{k - 1} ∣^{2} ∣ D_{n, k} ∣^{2} + 2 ∣ D_{n, k} ∣^{4}$ give $a_{k} \leq a_{k - 1} + 8 E (∣ S_{k - 1} ∣^{2} v_{k}) + 3 E w_{k} .$ By Cauchy–Schwarz and conditional Jensen, $E (∣ S_{k - 1} ∣^{2} v_{k}) \leq \sqrt{a_{k - 1}} \sqrt{E v_{k}^{2}} \leq \sqrt{a_{k - 1}} \sqrt{E w_{k}} \leq C \sqrt{a_{k - 1}} / N_{n}^{2}$ , while $E w_{k} \leq C / N_{n}^{4}$ . Induction gives $a_{k} \leq C_{0} k^{2} / N_{n}^{4}$ for a fixed sufficiently large $C_{0}$ and $k \leq N_{n}$ , hence $E ∣ S_{N_{n}} ∣^{4} \leq C_{0} / N_{n}^{2}$ .

5.1givenF5F7step 4.1algebra∎

Apply [F7] to $Y = ∣ S_{N_{n}} ∣^{4}$ and $a = t^{4}$ to obtain the stated $C / (t^{4} N_{n}^{2})$ tail bound. Since $N_{n} / n \to κ > 0$ , the probabilities are summable in $n$ . All deviations lie on the common probability space in the Statement, so [F5] proves almost-sure convergence for this fixed $z$ .

Source notes

Bai, Miao, and Pan, §2, equations (2.1)–(2.3), printed pp. 12–13 (full paper PDF).

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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)
J. W. Silverstein, Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices (standard reference, not scraped)