Pipeline-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.

Sample-covariance resolvent deterministic equivalents

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Directional spectral measure of a Hermitian matrix

Definition

For positive integers $d, N$ , let $X \in C^{d \times N}$ and let $T \in C^{d \times d}$ be Hermitian positive semidefinite, meaning $T = T^{*}$ and $v^{*} T v \geq 0$ for every $v \in C^{d}$ . Let $T^{1 / 2}$ be its Hermitian positive semidefinite square root and set $A : = \frac{1}{N} T^{1 / 2} X X^{*} T^{1 / 2} .$ For each distinct eigenvalue $λ$ of the Hermitian matrix $A$ , let $P_{λ}$ be the orthogonal projection onto its eigenspace. The empirical spectral distribution of $A$ , normalized by its dimension $d$ , is $μ_{A} : = \frac{1}{d} \sum_{λ \in spec (A)} rank (P_{λ}) δ_{λ},$ where $δ_{λ}$ is the unit point mass at $λ$ . For a deterministic unit vector $x \in C^{d}$ , its directional spectral measure is $μ_{A}^{x} : = \sum_{λ \in spec (A)} ∥ P_{λ} x ∥_{2}^{2} δ_{λ} .$ For a Borel set $B \subseteq R$ , set $1_{B} (A) : = \sum_{λ \in B \cap spec (A)} P_{λ}$ ; then $μ_{A}^{x} (B) = x^{*} 1_{B} (A) x$ . Both measures are supported on the nonnegative real axis.

Definition references

Bai, Miao, and Pan, §1, printed pp. 7–8 (full paper PDF).

Bai and Silverstein, Chapter 3, §§3.1–3.3, printed pp. 39–58 (full book PDF).

Facts & Assumptions

Given: Positive integers $d, N$ , the matrices and vector in the Definition, and the finite-dimensional spectral decomposition.

[F1]

Every Hermitian matrix has real eigenvalues and mutually orthogonal eigenspace projections whose sum is the identity (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F2]

For each real $λ$ , $δ_{λ}$ is the unit Borel point mass at $λ$ (The Dirac set function at a point).

[F3]

Each point mass $δ_{λ}$ is a Borel probability measure (A Dirac set function is a probability measure).

Verification

Given: The positive semidefinite matrix $T$ , matrix $X$ , and the definitions in the Definition.

1.1givenF1algebra

By [F1], diagonalize $T = U diag (t_{1}, \dots, t_{d}) U^{*}$ with each $t_{j} \geq 0$ . Then $T^{1 / 2} : = U diag (\sqrt{t_{1}}, \dots, \sqrt{t_{d}}) U^{*}$ is Hermitian positive semidefinite. With $B = N^{- 1 / 2} T^{1 / 2} X$ , the matrix $A = B B^{*}$ is Hermitian and $v^{*} A v = ∥ B^{*} v ∥_{2}^{2} \geq 0$ for every $v$ ; hence all its eigenvalues are nonnegative.

2.1givenF2F3step 1.1algebra∎

Each displayed set function is a finite nonnegative linear combination of the point masses defined in [F2], which are Borel probability measures by [F3]; its coefficients are nonnegative because $rank (P_{λ}) / d \geq 0$ and $∥ P_{λ} x ∥_{2}^{2} \geq 0$ . Thus both are finite Borel measures. By step 1.1 their finite supports lie in $[0, \infty)$ .

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

Operator-norm bound for bounded complex iid matrices

Statement

On a common probability space, for each positive integer $n$ , let $N_{n}$ be a positive integer with $n / N_{n} \to c \in (0, \infty)$ , and let $X_{n} \in C^{n \times N_{n}}$ have independent identically distributed entries $ξ$ satisfying $E ξ = 0$ and $∣ ξ ∣ \leq K$ almost surely for a fixed $K < \infty$ . Then there is a finite constant $C = C (K, c)$ such that, almost surely, $∥ X_{n} ∥_{o p} \leq C \sqrt{N_{n}}$ for all sufficiently large $n$ .

Statement references

Bai and Silverstein, Chapter 3, full book PDF (source).

Facts & Assumptions

Given: The matrix sequence on its common probability space and its bounded centered iid entry law from the Statement.

[F1]

The entries in each matrix are mutually independent and have the common law specified in the Statement (Identical distribution and IID families).

[F2]

If $\sum_{n} P (E_{n}) < \infty$ , then almost surely only finitely many $E_{n}$ occur (First Borel-Cantelli lemma for events).

[F3]

For an event $E$ and $a > 0$ , $P (E) \leq E [Y] / a$ whenever $Y \geq a 1_{E}$ (Markov's inequality for random variables).

[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

1.1givenF4algebra

If $K = 0$ , every entry is zero almost surely and the conclusion is immediate. Assume $K > 0$ . Regard the complex unit sphere in $C^{d}$ as the real unit sphere in $R^{2 d}$ . For $0 < ε < 1 / 2$ , a maximal $ε$ -separated subset is an $ε$ -net; the disjoint radius- $ε / 2$ balls about its points lie in the radius- $(1 + ε / 2)$ ball, so volume comparison bounds its cardinality by $(1 + 2 / ε)^{2 d}$ . If $N_{1}, N_{2}$ are such nets for the two unit spheres and $A$ is an $n \times N$ complex matrix, approximating unit vectors $u, v$ attaining $∥ A ∥_{o p} = ∣ u^{*} A v ∣$ gives $∥ A ∥_{o p} \leq (1 - 2 ε)^{- 1} \max_{u_{0} \in N_{1}, v_{0} \in N_{2}} ∣ u_{0}^{*} A v_{0} ∣ .$ Indeed, the approximation error is at most $∥ u - u_{0} ∥ ∥ A ∥ + ∥ v - v_{0} ∥ ∥ A ∥ \leq 2 ε ∥ A ∥$ .

2.1givenF1F3step 1.1algebra

Fix deterministic unit vectors $u \in C^{n}$ and $v \in C^{N}$ . Then $u^{*} X_{n} v = \sum_{i, j} {\overline{u}}_{i} v_{j} ξ_{i j}$ is a sum of independent centered complex variables. Each real or imaginary summand is bounded in absolute value by $K ∣ u_{i} ∣ ∣ v_{j} ∣$ . If a centered real variable $Y$ is bounded by $b$ , convexity of the exponential on $[- b, b]$ and $E Y = 0$ give $E e^{λ Y} \leq \cosh (λ b) \leq e^{λ^{2} b^{2} / 2}$ . Independence, exponential Markov inequality [F3], and optimization in $λ$ therefore give $P {∣ u^{*} X_{n} v ∣ > t} \leq 4 \exp ⁣ (- \frac{t^{2}}{4 K^{2}}),$ because $\sum_{i, j} K^{2} ∣ u_{i} ∣^{2} ∣ v_{j} ∣^{2} = K^{2}$ .

3.1givenF2step 1.1step 2.1algebra∎

Choose $ε = 1 / 8$ in step 1.1. The product of the two net cardinalities is at most $17^{2 (n + N_{n})}$ . Taking $t = L \sqrt{n + N_{n}}$ with $L$ large enough that $L^{2} / (4 K^{2}) > 2 \log 17 + 1$ , the union bound and step 2.1 make the probability that the net maximum exceeds $t$ at most $4 e^{- (n + N_{n})}$ . These probabilities are summable in $n$ because $N_{n} / n \to 1 / c > 0$ . Since all $X_{n}$ are defined on the common probability space in the Statement, [F2] applies to this sequence of events and the net maximum is at most $L \sqrt{n + N_{n}}$ eventually almost surely. The net inequality in step 1.1 and boundedness of $n / N_{n}$ then yield $∥ X_{n} ∥_{o p} \leq C \sqrt{N_{n}}$ eventually, with deterministic $C = C (K, c)$ .

Source notes

The proof is the explicit finite-net and exponential-moment argument above; the sources record the standard norm-estimation context and are not imported as proof premises.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Fourth-moment bound for a centered complex quadratic form

Statement

Let $ξ = (ξ_{1}, \dots, ξ_{d})^{T}$ have independent identically distributed complex coordinates with $E ξ_{i} = 0$ , $E ∣ ξ_{i} ∣^{2} = 1$ , and $μ_{4} : = E ∣ ξ_{i} ∣^{4} < \infty$ . Let $A$ be an independent random complex $d \times d$ matrix with $E ∥ A ∥_{F}^{2} < \infty$ , where $∥ A ∥_{F}^{2} : = \sum_{i, j} ∣ A_{i j} ∣^{2}$ . Put $τ : = E ξ_{1}^{2}$ . Then $E {∣ ξ^{*} A ξ - tr A ∣}^{2} \leq C_{μ_{4}, τ} E ∥ A ∥_{F}^{2}, C_{μ_{4}, τ} : = \max {μ_{4} - 1, 1 + ∣ τ ∣^{2}} .$

Statement references

Bai, Miao, and Pan, Lemma 1 in §2, printed p. 12 (full paper PDF).

Bai and Silverstein, Chapter 3, §3.3, printed pp. 57–58 (full book PDF).

Facts & Assumptions

Given: The independent vector and matrix, moment conditions, and Frobenius norm in the Statement.

[F1]

IID means the coordinates are mutually independent and have one common law (Identical distribution and IID families).

[F2]

The joint law of independent random elements is the product of their marginal laws (Independent random elements have product joint law).

[F3]

Fubini's theorem permits integration in either order for an integrable function on a sigma-finite product space (Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

Given: The moment hypotheses, independence, and finite Frobenius-norm second moment in the Statement.

1.1givenF1algebra

First fix a deterministic complex matrix $A$ . Centering the diagonal terms gives $ξ^{*} A ξ - tr A = \sum_{i} A_{i i} (∣ ξ_{i} ∣^{2} - 1) + \sum_{i \neq j} \overline{ξ_{i}} A_{i j} ξ_{j} .$ In the squared modulus, independence and $E ξ_{i} = 0$ make diagonal and off-diagonal parts uncorrelated; distinct diagonal terms are uncorrelated; and two off-diagonal terms have nonzero expectation only when their ordered index pairs agree or are reversed. Expanding those remaining terms yields $E ∣ ξ^{*} A ξ - tr A ∣^{2} = (μ_{4} - 1) \sum_{i} ∣ A_{i i} ∣^{2} + \sum_{i \neq j} ∣ A_{i j} ∣^{2} + ∣ τ ∣^{2} \sum_{i \neq j} A_{i j} \overline{A_{j i}} .$ Here $μ_{4} - 1 \geq 0$ by Cauchy–Schwarz, and $∣ τ ∣ \leq E ∣ ξ_{1} ∣^{2} = 1$ . Also $∣ \sum_{i \neq j} A_{i j} \overline{A_{j i}} ∣ \leq \sum_{i \neq j} ∣ A_{i j} ∣ ∣ A_{j i} ∣ \leq \sum_{i \neq j} ∣ A_{i j} ∣^{2} .$ Thus the deterministic quadratic-form variance is at most $C_{μ_{4}, τ} ∥ A ∥_{F}^{2}$ .

2.1givenF2F3step 1.1algebra∎

By [F2], $(ξ, A)$ has product joint law. The integrand is integrable because $∣ ξ^{*} A ξ - tr A ∣^{2} \leq 2 ∥ A ∥_{F}^{2} ∥ ξ ∥_{2}^{4} + 2 d ∥ A ∥_{F}^{2},$ and $E ∥ ξ ∥_{2}^{4} < \infty$ in finite dimension. Integrate the deterministic inequality from step 1.1 over the law of $A$ ; [F3] justifies the order of integration and gives the stated bound.

Source notes

Bai, Miao, and Pan, Lemma 1 in §2, printed p. 12 (full paper PDF).

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Uniqueness of the deformed Marchenko–Pastur fixed point

Statement

Let $c > 0$ , let $H$ be a Borel probability measure on $[0, \infty)$ , and fix $z \in C$ with $Im z > 0$ . There is at most one $q \in C$ satisfying $Im q > 0$ , $u : = - (1 - c) / z + c q$ with $Im u > 0$ , and $q = \int_{[0, \infty)} \frac{1}{t (1 - c - c z q) - z} d H (t) .$

Statement references

Silverstein, Theorem 1.1 and equation (1.4), PDF pp. 2–3 (full paper PDF).

Bai, Miao, and Pan, equation (1.1), PDF pp. 3–4 (full paper PDF).

Facts & Assumptions

Given: $c > 0$ , a Borel probability measure $H$ on $[0, \infty)$ , $z \in C^{+}$ , and two candidate solutions $q_{1}, q_{2}$ satisfying the Statement.

[A1]

The aspect ratio $c$ is positive.

[A2]

$H$ is a countably additive Borel measure on $[0, \infty)$ with total mass one (Probability measures and probability spaces).

[A3]

The spectral parameter is $z = u + i η$ with $u \in R$ and $η > 0$ .

[A4]

Each candidate $q$ has positive imaginary part, $- (1 - c) / z + c q$ has positive imaginary part, and $q$ satisfies the fixed-point equation in the Statement.

Proof

1.1givenA1A2A3A4algebra

For either candidate $q$ , set $u = - (1 - c) / z + c q \in C^{+}$ . Since $t \geq 0$ , $1 + t u \neq 0$ for every $t$ , and the fixed-point equation becomes $q = - z^{- 1} \int (1 + t u)^{- 1} d H (t)$ . Substituting this into $u = - (1 - c) / z + c q$ and using $(1 + t u)^{- 1} = 1 - t u / (1 + t u)$ gives $z = - \frac{1}{u} + c \int_{[0, \infty)} \frac{t}{1 + t u} d H (t) .$ All integrals are finite because $∣ t / (1 + t u) ∣ \leq 1 / Im u$ .

2.1A1A2A3A4step 1.1algebra

If $u_{1} \neq u_{2}$ are the transforms of two solutions, subtracting their equations in step 1.1 and dividing by $u_{1} - u_{2}$ yields $\frac{1}{u_{1} u_{2}} = c \int_{[0, \infty)} \frac{t^{2}}{(1 + t u_{1}) (1 + t u_{2})} d H (t) .$ For either solution $u$ , taking imaginary parts in step 1.1 gives $Im z = Im u (\frac{1}{∣ u ∣^{2}} - c \int_{[0, \infty)} \frac{t^{2}}{∣ 1 + t u ∣^{2}} d H (t)) > 0,$ so $c ∣ u ∣^{2} \int t^{2} ∣ 1 + t u ∣^{- 2} d H (t) < 1$ .

3.1A1A2A3A4step 2.1algebra∎

Taking absolute values in the identity of step 2.1 and applying Cauchy–Schwarz gives $1 \leq {(c ∣ u_{1} ∣^{2} \int \frac{t^{2}}{∣ 1 + t u_{1} ∣^{2}} d H (t))}^{1 / 2} {(c ∣ u_{2} ∣^{2} \int \frac{t^{2}}{∣ 1 + t u_{2} ∣^{2}} d H (t))}^{1 / 2} < 1,$ contradicting step 2.1. Hence $u_{1} = u_{2}$ , and the relation $u = - (1 - c) / z + c q$ with $c > 0$ gives $q_{1} = q_{2}$ .

Source notes

Silverstein, Theorem 1.1 and equation (1.4), PDF pp. 2–3 (full paper PDF).

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Hermitian rank-one resolvent update and trace-norm bound

Statement

Let $B \in C^{d \times d}$ be Hermitian positive semidefinite, let $x \in C^{d}$ , and let $z = u + i η$ with $u \in R$ and $η > 0$ . Set $R = (B - z I)^{- 1}$ and $R_{x} = (B + x x^{*} - z I)^{- 1}$ . For the trace norm $∥ M ∥_{1}$ , defined as the sum of the singular values of $M$ , one has $1 + x^{*} R x \neq 0$ , $R_{x} = R - \frac{R x x^{*} R}{1 + x^{*} R x}, ∥ R_{x} - R ∥_{1} \leq \frac{1}{η}, ∣ tr R_{x} - tr R ∣ \leq \frac{1}{η} .$

Statement references

Bai, Miao, and Pan, the rank-one identity in §2, printed pp. 12–13 (full paper PDF).

Bai and Silverstein, Chapter 3, §3.3, printed pp. 53–54 (full book PDF).

Facts & Assumptions

Given: The Hermitian positive semidefinite matrix $B$ , vector $x$ , and $z$ in the Statement.

[F1]

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

Proof

Given: $B$ , $x$ , and $z = u + i η$ with $η > 0$ .

1.1givenF1algebra

By [F1], write $B = U diag (λ_{1}, \dots, λ_{d}) U^{*}$ with $λ_{j} \geq 0$ . If $c = U^{*} x$ , then $q : = x^{*} R x = \sum_{j} \frac{∣ c_{j} ∣^{2}}{λ_{j} - z}, Im q = η \sum_{j} \frac{∣ c_{j} ∣^{2}}{∣ λ_{j} - z ∣^{2}} = η ∥ R x ∥_{2}^{2} .$ For $x \neq 0$ this is positive, so $1 + q \neq 0$ ; for $x = 0$ the denominator is one.

2.1step 1.1algebra

Direct multiplication, using step 1.1, gives $(B + x x^{*} - z I) (R - \frac{R x x^{*} R}{1 + x^{*} R x}) = I .$ The Hermitian matrix $B + x x^{*}$ has a resolvent at nonreal $z$ , so the displayed inverse is $R_{x}$ .

3.1step 1.1step 2.1algebra∎

The difference in step 2.1 is a rank-one matrix. Its trace norm equals the product of the two vector norms divided by $∣ 1 + q ∣$ ; normality of $R$ makes both vector norms $∥ R x ∥_{2}$ . Since $∣ 1 + q ∣ \geq Im q = η ∥ R x ∥_{2}^{2}$ , step 1.1 gives $∥ R_{x} - R ∥_{1} \leq 1 / η$ . The inequality $∣ tr M ∣ \leq ∥ M ∥_{1}$ then gives the trace bound. The case $x = 0$ has both differences zero.

Source notes

Bai, Miao, and Pan, §2, equations (2.1)–(2.2), printed pp. 12–13 (full paper PDF).

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

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).

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Concentration of the normalized complex Hermitian resolvent trace

Statement

On a common probability space, for each positive integer $n$ , let $Y_{n, 1}, \dots, Y_{n, N_{n}}$ be independent random vectors in $C^{n}$ , let $B_{n} = \sum_{j = 1}^{N_{n}} Y_{n, j} Y_{n, j}^{*}$ , and assume $N_{n} / n \to κ \in (0, \infty)$ . Fix $z = u + i η$ with $u \in R$ and $η > 0$ , and define $M_{n} (z) : = \frac{1}{n} tr (B_{n} - z I)^{- 1} .$ Then for every $t > 0$ , $P {∣ M_{n} (z) - E M_{n} (z) ∣ > t} \leq 4 \exp ⁣ (- \frac{t^{2} n^{2} η^{2}}{4 N_{n}}),$ and $M_{n} (z) - E M_{n} (z) \to 0$ almost surely.

Statement references

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

Bai and Silverstein, Chapter 3, §3.3, printed pp. 53–55 (full book PDF).

Facts & Assumptions

Given: The common probability space, independent complex columns $Y_{n, j}$ , the normalized resolvent trace, and $z = u + i η$ with $η > 0$ .

[F1]

Adding one positive rank-one term changes the unnormalized Hermitian resolvent trace by at most $1 / η$ (Hermitian rank-one resolvent update and trace-norm bound).

[F2]

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

[F3]

Fubini's theorem permits integration in either order on the finite product spaces used to define the conditional averages (Fubini's theorem for L^1 functions on a sigma-finite product).

[F4]

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

[F5]

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

Proof

Given: The hypotheses and notation of the Statement.

1.1givenF1algebra

Replacing one column $Y_{n, j}$ by any other vector removes one positive rank-one term and adds another. Applying [F1] twice shows that $tr (B_{n} - z I)^{- 1}$ changes by at most $2 / η$ , so the real and imaginary parts of $M_{n} (z)$ each have coordinate oscillation at most $c_{n} : = 2 / (n η)$ .

2.1givenF2F3step 1.1construct

Let $f$ be either real part of $M_{n} (z)$ and use [F2]–[F3] to form its product-integral Doob averages $F_{j}$ over the first $j$ columns, integrating the remaining columns against their marginal laws. Boundedness $∣ tr (B_{n} - z I)^{- 1} ∣ \leq n / η$ makes these integrals finite. Then $D_{j} : = F_{j} - F_{j - 1}$ has conditional mean zero and lies, conditionally on the first $j - 1$ columns, in an interval of length at most $c_{n}$ by step 1.1.

3.1F5step 2.1algebra

If a real random variable $D$ has mean zero and lies in an interval of width $c$ , then $\log E e^{λ D}$ has second derivative equal to a variance under an exponentially tilted law, at most $c^{2} / 4$ ; the variance bound follows from $E [(D - a) (b - D)] \geq 0$ for interval endpoints $a \leq D \leq b$ . Since the log moment and its first derivative vanish at $0$ , integrating twice gives $E e^{λ D} \leq e^{λ^{2} c^{2} / 8}$ . Apply this conditional estimate to each $D_{j}$ , iterate, and use [F5] with $λ = 4 t / (N_{n} c_{n}^{2})$ to obtain each one-sided tail $P {f - E f \geq t} \leq e^{- 2 t^{2} / (N_{n} c_{n}^{2})}$ . The same bound applies to the lower tail and to the imaginary part. If a complex number has modulus greater than $t$ , one of its real or imaginary parts has magnitude greater than $t / \sqrt{2}$ , so the union bound and $c_{n} = 2 / (n η)$ give the displayed probability inequality.

4.1givenF4step 3.1algebra∎

Since $N_{n} / n \to κ \in (0, \infty)$ , for every fixed $t > 0$ the step 3.1 bound is at most $4 e^{- a_{t} n}$ for all sufficiently large $n$ and some $a_{t} > 0$ . The probabilities are summable. The deviations lie on the common probability space in the Statement, so [F4] applies; use $t = 1 / m$ , $m \geq 1$ , and intersect the resulting countable probability-one events to conclude $M_{n} (z) - E M_{n} (z) \to 0$ almost surely.

Source notes

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

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

A Hermitian directional spectral measure is a probability law

Statement

For every positive integer $d$ and every Hermitian positive semidefinite $A \in C^{d \times d}$ , its empirical spectral distribution $μ_{A}$ is a Borel probability measure supported on $[0, \infty)$ . For every deterministic unit vector $x \in C^{d}$ , its directional spectral measure $μ_{A}^{x}$ is also a Borel probability measure supported on $[0, \infty)$ .

Statement references

Bai, Miao, and Pan, §1, printed pp. 7–8 (full paper PDF).

Bai and Silverstein, Appendix B.2, printed pp. 514–523 (full book PDF).

Directional and empirical spectral-measure definitions: definition of the directional spectral measure.

Facts & Assumptions

Given: A positive integer $d$ , a Hermitian positive semidefinite matrix $A \in C^{d \times d}$ and, for the directional measure, a deterministic vector $x \in C^{d}$ with $∥ x ∥_{2} = 1$ .

[F1]

A finite-dimensional Hermitian matrix has an orthonormal eigenbasis; its eigenvalues are real, and positive semidefiniteness makes them nonnegative (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F2]

The empirical and directional measures are the finite Dirac sums stated in Directional spectral measure of a Hermitian matrix.

[F3]

A Borel probability measure is countably additive and has total mass one (Probability measures and probability spaces).

[F4]

The Dirac set function $δ_{λ}$ defined in The Dirac set function at a point is a Borel probability measure by A Dirac set function is a probability measure.

Proof

Given: $A$ , $x$ , and the definitions in the Statement.

1.1givenF1F2F3F4algebra

By [F1], write $A = \sum_{j = 1}^{r} λ_{j} P_{j}$ over its distinct eigenvalues, where the $P_{j}$ are mutually orthogonal projections, $\sum_{j} P_{j} = I$ , and $λ_{j} \geq 0$ . Taking traces gives $\sum_{j} rank (P_{j}) = d$ . Thus the nonnegative weights $rank (P_{j}) / d$ in [F2] sum to one; by [F3] and [F4], $μ_{A}$ is a Borel probability measure supported on $[0, \infty)$ .

2.1givenF1F2F3F4step 1.1algebra∎

For the unit vector $x$ , the directional weights in [F2] are $∥ P_{j} x ∥_{2}^{2} \geq 0$ , and orthogonality gives $\sum_{j} ∥ P_{j} x ∥_{2}^{2} = ∥ \sum_{j} P_{j} x ∥_{2}^{2} = ∥ x ∥_{2}^{2} = 1$ . By [F3] and [F4], the finite Dirac sum is a Borel probability measure; its support is contained in the nonnegative spectrum of $A$ by [F1].

Source notes

Bai, Miao, and Pan, equation (1.8), printed p. 8 (full paper PDF).

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Deterministic equivalent for the bounded complex sample-covariance resolvent

Statement

For each positive integer $n$ , let $N_{n}$ be a positive integer with $N_{n} / n \to κ \in (0, \infty)$ , let $X_{n} \in C^{n \times N_{n}}$ have entries from one common iid law with mean zero, second absolute moment one and $∣ X_{i j} ∣ \leq K$ almost surely for a fixed $K$ , and let $T_{n}$ be deterministic Hermitian positive semidefinite (self-adjoint with $v^{*} T_{n} v \geq 0$ for every $v$ ) with induced Euclidean operator norm $∥ T_{n} ∥_{o p} \leq M$ . Define $A_{n} = N_{n}^{- 1} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}, {\tilde{A}}_{n} = N_{n}^{- 1} X_{n}^{*} T_{n} X_{n},$ $R_{n} (z) = (A_{n} - z I_{n})^{- 1}, {\tilde{R}}_{n} (z) = ({\tilde{A}}_{n} - z I_{N_{n}})^{- 1}, q_{n} (z) = E [N_{n}^{- 1} tr {\tilde{R}}_{n} (z)] .$ For every fixed $z = u + i η$ with $u \in R$ and $η > 0$ , the matrix $I_{n} + q_{n} (z) T_{n}$ is invertible and $\sup_{∥ x ∥_{2} = 1} ∣ x^{*} (E R_{n} (z) + \frac{1}{z} (I_{n} + q_{n} (z) T_{n})^{- 1}) x ∣ ⟶ 0.$

Statement references

Bai, Miao, and Pan, complete Section 2 proof of Theorem 1, printed pp. 12–16 (full paper PDF).

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

Facts & Assumptions

Given: The iid bounded entries, deterministic positive semidefinite $T_{n}$ , aspect-ratio limit, and fixed $z \in C^{+}$ in the Statement.

[F1]

Independent centered complex coordinates with unit second absolute moment satisfy the stated second-moment quadratic-form bound, also conditionally when the coefficient matrix is independent (Fourth-moment bound for a centered complex quadratic form).

[F2]

Adding one positive rank-one matrix has the Sherman–Morrison resolvent formula and trace-norm bound (Hermitian rank-one resolvent update and trace-norm bound).

[F3]

For bounded iid entries, each deterministic unit-vector resolvent quadratic form differs from its expectation with fourth-tail bound $C / (t^{4} N_{n}^{2})$ (Adapted concentration of bounded complex resolvent quadratic forms).

[F4]

Hermitian positive semidefinite matrices have nonnegative eigenvalues and an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F5]

The Stieltjes transform of a probability measure uses the $(t - z)^{- 1}$ convention in the upper half-plane (Stieltjes transform of a probability measure on the real line).

[F6]

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

1.1givenF4F5F6algebra

Fix $n, z$ , suppress the index $n$ , write $N = N_{n}$ , $s_{j} = T^{1 / 2} X_{\cdot j} / \sqrt{N}$ , $R = (A - z I)^{- 1}$ , $R_{j} = (A - s_{j} s_{j}^{*} - z I)^{- 1}$ , and $β_{j} = (1 + s_{j}^{*} R_{j} s_{j})^{- 1}$ . The companion matrix $\tilde{A} = N^{- 1} X^{*} T X$ is positive semidefinite by [F4], so $q = N^{- 1} E tr (\tilde{A} - z I)^{- 1}$ is the Stieltjes transform of an expected probability measure on $[0, \infty)$ by [F5] and has positive imaginary part. If $q = \int (λ - z)^{- 1} d G (λ)$ , then for every $t \geq 0$ , $Im {z (1 + t q)} = η + t \int \frac{λ η}{∣ λ - z ∣^{2}} d G (λ) \geq η,$ so $C : = (I + q T)^{- 1}$ exists and $∥ C ∥_{o p} \leq ∣ z ∣ / η$ . The same spectral calculation applied to $A - s_{j} s_{j}^{*}$ gives $∣ β_{j} ∣ \leq ∣ z ∣ / η$ .

2.1givenF1F4step 1.1algebra

Set $b_{j} = (1 + N^{- 1} tr (T R_{j}))^{- 1}$ and $ξ_{j} = s_{j}^{*} R_{j} s_{j} - N^{- 1} tr (T R_{j})$ . The trace in $b_{j}$ is a Stieltjes transform with nonnegative weights: in an eigenbasis of $A - s_{j} s_{j}^{*}$ its weights are $⟨ v_{k}, T v_{k} ⟩ / N \geq 0$ . The inequality in step 1.1 therefore also gives $∣ b_{j} ∣ \leq ∣ z ∣ / η$ . Conditional on all columns except $j$ , [F1] and $∥ T^{1 / 2} R_{j} T^{1 / 2} ∥_{F}^{2} \leq n M^{2} / η^{2}$ give $E_{j} ∣ ξ_{j} ∣^{2} \leq C_{1} n / N^{2} = O (N^{- 1})$ , where $E_{j}$ integrates column $j$ . For any deterministic unit $x$ , set $α_{j} = x^{*} C s_{j} s_{j}^{*} R_{j} x - N^{- 1} x^{*} C T R_{j} x .$ This is a centered quadratic form in column $j$ . Its coefficient matrix is $N^{- 1} (T^{1 / 2} R_{j} x) (x^{*} C T^{1 / 2}),$ so it has rank at most one and Frobenius norm at most $M ∥ C ∥_{o p} / (N η)$ . Thus [F1] gives $E_{j} ∣ α_{j} ∣^{2} = O (N^{- 2})$ . Also $E_{j} α_{j} = 0$ and $β_{j} - b_{j} = - β_{j} b_{j} ξ_{j}$ .

2.2givenF4step 1.1algebra

For all sufficiently large $n$ we have $N = N_{n} \geq 2$ . Put $Y = [s_{2} \dots s_{N}]$ , $D = Y^{*} Y - z I_{N - 1}$ , and $C_{-} = Y Y^{*} - z I_{n} = A - s_{1} s_{1}^{*} - z I_{n}$ . The matrix $\tilde{A} - z I$ has block form $(\begin{array}{cc} s_{1}^{*} s_{1} - z & s_{1}^{*} Y \\ Y^{*} s_{1} & D \end{array}) .$ For a block matrix $(\begin{array}{cc} a & b^{*} \\ b & D \end{array})$ with invertible $D$ , solving its product with $(u, v)^{T} = (1, 0)^{T}$ gives $v = - D^{- 1} b u$ and $(a - b^{*} D^{- 1} b) u = 1$ . Hence its inverse's upper-left entry is $(a - b^{*} D^{- 1} b)^{- 1}$ . Here $(C_{-}) Y = Y D$ , so $(C_{-}) (I - Y D^{- 1} Y^{*}) = C_{-} - Y Y^{*} = - z I$ and $I - Y D^{- 1} Y^{*} = - z C_{-}^{- 1} = - z R_{1}$ . The Schur complement is therefore $s_{1}^{*} (I - Y D^{- 1} Y^{*}) s_{1} - z = - z (1 + s_{1}^{*} R_{1} s_{1})$ , proving $(\tilde{R})_{11} = - β_{1} / z$ .

3.1givenF2F4step 1.1step 2.2algebra

By exchangeability of the columns and step 2.2, $z q = - N^{- 1} \sum_{j = 1}^{N} E β_{j}$ . The identity $R^{- 1} = \sum_{j} s_{j} s_{j}^{*} - z I$ gives $\sum_{j} s_{j} s_{j}^{*} R = I + z R$ . The rank-one formula [F2] gives $s_{j} s_{j}^{*} R = β_{j} s_{j} s_{j}^{*} R_{j}$ . Substituting these identities and the formula for $z q$ yields $E R + \frac{1}{z} C = \frac{1}{z} \sum_{j = 1}^{N} E [β_{j} (C s_{j} s_{j}^{*} R_{j} - N^{- 1} C T E R)] .$ Testing against $x$ and adding then subtracting $N^{- 1} x^{*} C T R_{j} x$ and $N^{- 1} x^{*} C T R x$ decomposes the right side into three terms $δ_{1} + δ_{2} + δ_{3}$ .

4.1givenF1step 2.1step 3.1algebra

For the first term, $δ_{1} = (N / z) E (β_{1} α_{1})$ . Since $b_{1}$ is independent of column 1 and $E_{1} α_{1} = 0$ , we have $E (b_{1} α_{1}) = 0$ . Therefore step 2.1, Cauchy–Schwarz, and the uniform bounds on $β_{1}, b_{1}$ imply $∣ δ_{1} ∣ \leq C_{2} N (E ∣ ξ_{1} ∣^{2})^{1 / 2} (E ∣ α_{1} ∣^{2})^{1 / 2} = O (N^{- 1 / 2}) .$

4.2givenF1F2step 2.1step 3.1algebra

For the second term, $R_{1} - R = β_{1} R_{1} s_{1} s_{1}^{*} R_{1}$ . Conditional second moments of the two linear forms $x^{*} C T R_{1} s_{1}$ and $s_{1}^{*} R_{1} x$ are each $O (N^{- 1})$ , uniformly in the other columns. Cauchy–Schwarz and boundedness of $β_{1}$ yield $∣ δ_{2} ∣ = O (N^{- 1})$ .

4.3givenF3step 1.1step 3.1algebra

The third term is $δ_{3} = z^{- 1} E [β_{1} x^{*} C T (R - E R) x]$ . For any deterministic sequence of unit vectors $x = x_{n}$ , put $u = (C T)^{*} x$ and $v = x$ ; their norms are uniformly bounded by step 1.1. For $Q (w) = w^{*} (R - E R) w$ , direct expansion gives $B_{n} : = u^{*} (R - E R) v = \frac{1}{4} {Q (u + v) - Q (u - v) - i Q (u + i v) + i Q (u - i v)} .$ Rescale each nonzero vector to unit length; zero vectors contribute zero. For each fixed $t > 0$ , [F3] and the union bound imply $P (∣ B_{n} ∣ > t) \to 0$ . Also $∣ B_{n} ∣ \leq 2 ∥ u ∥ ∥ v ∥ / η \leq L$ for a fixed finite $L$ . Thus $E ∣ B_{n} ∣^{2} \leq t^{2} + L^{2} P (∣ B_{n} ∣ > t)$ , so first letting $n \to \infty$ and then $t ↓ 0$ gives $E ∣ B_{n} ∣^{2} \to 0$ . Boundedness of $β_{1}$ and Cauchy–Schwarz yield $δ_{3} \to 0$ . This holds for every deterministic unit-vector sequence, hence uniformly over unit $x$ : otherwise a sequence of deterministic directions with $∣ δ_{3} ∣$ bounded away from zero along a subsequence would contradict the conclusion just proved. No shrinking-threshold tail estimate is used.

5.1givenstep 3.1step 4.1step 4.2step 4.3algebra∎

The bounds in steps 4.1, 4.2, and 4.3 tend to zero uniformly over deterministic unit $x$ . The tested resolvent identity in step 3.1 therefore proves the asserted supremum limit.

Source notes

Bai, Miao, and Pan, equations (2.2)–(2.9), printed pp. 12–15 (full paper PDF). The $N$ -prefactor in their estimate (2.6) gives $O (N^{- 1 / 2})$ after the displayed moment bounds; the proof above retains that corrected vanishing order. Their (2.8) term is bounded here by adapted resolvent concentration and polarization.

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

General-population sample-covariance law for bounded entries

Statement

Assume the Axiom of Choice. On a common probability space, for each positive integer $n$ , let $N_{n}$ be a positive integer with $N_{n} / n \to κ \in (0, \infty)$ , let $X_{n} \in C^{n \times N_{n}}$ have entries from one common iid law with mean zero, second absolute moment one and absolute value bounded by one fixed finite constant, and let $T_{n}$ be deterministic Hermitian positive semidefinite (self-adjoint with $v^{*} T_{n} v \geq 0$ for every $v$ ) with uniformly bounded induced Euclidean operator norm. Suppose the empirical eigenvalue measures $H_{n} = n^{- 1} \sum_{i = 1}^{n} δ_{λ_{i} (T_{n})}$ converge weakly to a Borel probability measure $H$ on $[0, \infty)$ . Put $c = κ^{- 1}$ and $A_{n} = N_{n}^{- 1} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}$ , and let $F_{n} = n^{- 1} \sum_{i = 1}^{n} δ_{λ_{i} (A_{n})}$ . Then there is a unique Borel probability measure $F_{c, H}$ on $[0, \infty)$ such that, almost surely, $F_{n}$ converges weakly to $F_{c, H}$ . Its Stieltjes transform $m_{c, H} (z) = \int (t - z)^{- 1} d F_{c, H} (t)$ is the unique $m \in C$ satisfying $Im m > 0$ , $u : = - (1 - c) / z + c m$ with $Im u > 0$ , and $m = \int_{[0, \infty)} \frac{1}{t (1 - c - c z m) - z} d H (t) (Im z > 0) .$

Statement references

Silverstein, Theorem 1.1 and its complete proof in Section 3 (full paper PDF).

Bai, Miao, and Pan, Section 2, including the sample-covariance resolvent conventions (full paper PDF).

Facts & Assumptions

Given: The common probability space, assumptions, matrices, and empirical measures in the Statement, including AC.

[A1]

The Axiom of Choice permits a choice function for every family of nonempty sets (The Axiom of Choice).

[F1]

The normalized eigenvalue measure of the complex Hermitian sample-covariance matrix $A_{n}$ is its empirical spectral distribution, defined as the rank-weighted finite spectral measure (Directional spectral measure of a Hermitian matrix).

[F2]

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

[F3]

The Stieltjes transform convention is $m_{μ} (z) = \int (t - z)^{- 1} d μ (t)$ for $Im z > 0$ (Stieltjes transform of a probability measure on the real line).

[F4]

Markov's inequality bounds the mass of ${t > R}$ by the first moment divided by $R$ (Markov's inequality for random variables).

[F5]

Tight families of Borel probabilities on a Polish space are relatively sequentially compact for weak convergence, under AC (Prokhorov tightness theorem on polish spaces).

[F6]

The bounded-entry resolvent has the deterministic equivalent $- (z (I + q_{n} T_{n}))^{- 1}$ , uniformly on deterministic unit-vector quadratic forms (Deterministic equivalent for the bounded complex sample-covariance resolvent).

[F7]

The normalized resolvent trace concentrates almost surely about its expectation (Concentration of the normalized complex Hermitian resolvent trace).

[F8]

The deformed fixed-point equation has at most one solution satisfying the stated upper-half-plane conditions (Uniqueness of the deformed Marchenko–Pastur fixed point).

[F9]

Equal Stieltjes transforms determine probability measures on the real line (Stieltjes inversion and uniqueness for probability measures on the real line).

[F10]

The normalized operator norm of bounded complex iid matrices is almost surely eventually bounded (Operator-norm bound for bounded complex iid matrices).

[F11]

The nonzero eigenvalues of $Y^{*} Y$ and $Y Y^{*}$ agree with multiplicity (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F12]

A probability measure is a countably additive Borel measure of total mass one (Probability measures and probability spaces).

[F13]

Integrals of nonnegative increasing measurable sequences converge to the integral of their pointwise limit (Monotone convergence for the integral).

[F14]

The ordered eigenvalues of a Hermitian matrix vary continuously with the matrix in operator norm (Ordered eigenvalues of Hermitian matrices are operator-norm Lipschitz).

[F15]

A Polish space is a separable topological space whose topology is induced by a complete metric (Polish spaces are separable completely metrizable spaces).

[F16]

Weak convergence of Borel probabilities means convergence of integrals against all bounded continuous real-valued functions (Weak convergence of borel probability measures).

[F17]

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

1.1givenA1F1F2F4F5F12F13F14F15F17algebra

Write $c_{n} = n / N_{n}$ , let ${\overset{ˉ}{F}}_{n} (B) = E F_{n} (B)$ , and let ${\overset{ˉ}{G}}_{n}$ be defined similarly for ${\tilde{A}}_{n} = N_{n}^{- 1} X_{n}^{*} T_{n} X_{n}$ . By [F14], each ordered eigenvalue is continuous in the matrix entries, so these setwise expectations are defined for Borel $B$ . For disjoint Borel sets, countable additivity of the random ESDs followed by [F13] gives countable additivity of ${\overset{ˉ}{F}}_{n}$ and ${\overset{ˉ}{G}}_{n}$ ; their total masses are one. Independence and unit second moment give $\int t d {\overset{ˉ}{F}}_{n} (t) = n^{- 1} tr T_{n}$ and $\int t d {\overset{ˉ}{G}}_{n} (t) = N_{n}^{- 1} tr T_{n}$ . Both first moments are uniformly bounded because $c_{n} \to c > 0$ and $∥ T_{n} ∥_{o p}$ is uniformly bounded. By [F4], their tails above $R$ are at most the corresponding first moment divided by $R$ , so both expected-measure sequences are tight on $[0, \infty)$ . The half-line is Polish: it is a closed subset of the complete real line and $Q \cap [0, \infty)$ is dense. By [A1] and [F5], every subsequence of ${\overset{ˉ}{F}}_{n}$ has a further weakly convergent subsequence.

1.2givenF7

For each fixed $z \in C^{+}$ , [F7] gives $\int (t - z)^{- 1} d F_{n} (t) - \int (t - z)^{- 1} d {\overset{ˉ}{F}}_{n} (t) \to 0$ almost surely. Since all $X_{n}$ are on the common probability space in the Statement, intersect these events over the countable dense set ${a + i b : a, b \in Q, b > 0}$ .

2.1givenF3F4F11F16step 1.1algebra

Fix $z \in C^{+}$ and take any such subsequence with ${\overset{ˉ}{F}}_{n} \Rightarrow G$ . Let $m_{G} (z) = \int (t - z)^{- 1} d G (t)$ and $q_{n} (z) = \int (t - z)^{- 1} d {\overset{ˉ}{G}}_{n} (t)$ . By [F11], the two sample-covariance matrices have the same nonzero eigenvalues, so their transforms satisfy exactly $q_{n} = - (1 - c_{n}) / z + c_{n} \int (t - z)^{- 1} d {\overset{ˉ}{F}}_{n} (t)$ . Hence $q_{n} \to u : = - (1 - c) / z + c m_{G} (z)$ . Moreover $Im m_{G} (z) > 0$ . The first moments of ${\overset{ˉ}{G}}_{n}$ are bounded, so at least half of each ${\overset{ˉ}{G}}_{n}$ lies in some fixed interval $[0, L]$ ; consequently $Im q_{n} (z) \geq η / (2 (L + ∣ z ∣)^{2})$ and $Im u > 0$ .

3.1givenF3F6F8F9F16F17step 1.1step 2.1algebra

Apply [F6] to an orthonormal eigenbasis of $T_{n}$ . Its uniform quadratic-form error implies $\int (t - z)^{- 1} d {\overset{ˉ}{F}}_{n} (t) = - \frac{1}{z n} tr (I + q_{n} T_{n})^{- 1} + o (1) = - \frac{1}{z} \int \frac{1}{1 + q_{n} t} d H_{n} (t) + o (1) .$ Since $H_{n} \Rightarrow H$ , all $H_{n}$ are supported in one compact interval, $q_{n} \to u$ , and $Im u > 0$ , the integrands converge uniformly on that interval to $- 1 / [z (1 + u t)]$ . Thus $m_{G} (z) = \int [- z (1 + u t)]^{- 1} d H (t)$ . The relation for $u$ in step 2.1 gives $- z (1 + u t) = t (1 - c - c z m_{G} (z)) - z$ , so $m_{G}$ satisfies the fixed-point equation in the Statement. The argument applies to every $z \in C^{+}$ and every weak subsequential limit $G$ . By [F8], all such limits have the same transform; [F9] makes them equal. At least one such limit exists by step 1.1, so call it $F_{c, H}$ . Tightness then implies ${\overset{ˉ}{F}}_{n} \Rightarrow F_{c, H}$ .

4.1givenA1F5F9F10F15F16step 1.1step 3.1step 1.2algebra∎

On the same event, [F10] gives $∥ A_{n} ∥_{o p} \leq L_{1}$ eventually for a finite deterministic $L_{1}$ , so the random measures $F_{n}$ are eventually supported on $[0, L_{1}]$ and are tight. The half-line is Polish by the metric facts in step 1.1 and [F15]. By [A1] and [F5], every subsequence has a weakly convergent further subsequence. For each dense-set $z$ , its transform equals that of $F_{c, H}$ by steps 3.1 and 1.2, since the real and imaginary parts of $(t - z)^{- 1}$ are bounded continuous and [F16] applies to each. For any probability law $μ$ , its transform satisfies $∣ m_{μ} (z) - m_{μ} (w) ∣ \leq ∣ z - w ∣ / (Im z Im w)$ , so equality on the dense set extends to all $C^{+}$ ; [F9] identifies the subsequential limit as $F_{c, H}$ . Every subsequence has the same further limit, and [F16] gives $F_{n} \Rightarrow F_{c, H}$ almost surely.

Source notes

Silverstein, Theorem 1.1 and complete Section 3 proof (full paper PDF). The local proof makes the companion-transform relation and the uniqueness step explicit.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Directional deterministic equivalent for bounded sample-covariance entries

Statement

Assume the Axiom of Choice. On a common probability space, for each positive integer $n$ , let $N_{n}$ be a positive integer with $N_{n} / n \to κ \in (0, \infty)$ , let $X_{n} \in C^{n \times N_{n}}$ have entries from one common iid law with mean zero, second absolute moment one and absolute value bounded by a fixed finite constant, and let $T_{n}$ be deterministic Hermitian positive semidefinite (self-adjoint with $v^{*} T_{n} v \geq 0$ for every $v$ ) with uniformly bounded induced Euclidean operator norm and empirical eigenvalue measures $H_{n} = n^{- 1} \sum_{i} δ_{λ_{i} (T_{n})} \Rightarrow H$ . Let $x_{n} \in C^{n}$ be deterministic unit vectors such that for every $w \in C ∖ [0, \infty)$ , $x_{n}^{*} (T_{n} - w I_{n})^{- 1} x_{n} ⟶ m_{H} (w) : = \int_{[0, \infty)} (t - w)^{- 1} d H (t) .$ For $A_{n} = N_{n}^{- 1} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}$ , define the directional spectral measure $μ_{n}^{x_{n}} (B) = x_{n}^{*} 1_{B} (A_{n}) x_{n}$ for Borel $B \subseteq [0, \infty)$ . Then almost surely $μ_{n}^{x_{n}} \Rightarrow F_{c, H}$ , where $c = κ^{- 1}$ and $F_{c, H}$ is the probability measure characterized by $m_{c, H} (z) = \int_{[0, \infty)} \frac{1}{t (1 - c - c z m_{c, H} (z)) - z} d H (t), Im z > 0.$

Statement references

Bai, Miao, and Pan, Theorem 1 and complete Section 2 proof, printed pp. 8–16 (full paper PDF).

Silverstein, Theorem 1.1 and complete Section 3 proof (full paper PDF).

Facts & Assumptions

Given: The common probability space, hypotheses, matrices, and directional measures in the Statement, including AC.

[A1]

The Axiom of Choice permits a choice function for every family of nonempty sets (The Axiom of Choice).

[F1]

For Hermitian $A_{n}$ and unit $x_{n}$ , the directional spectral measure is $\sum_{λ} ∥ P_{λ} x_{n} ∥_{2}^{2} δ_{λ}$ (Directional spectral measure of a Hermitian matrix).

[F2]

This measure is a Borel probability because the spectral projections are orthogonal and sum to the identity (A Hermitian directional spectral measure is a probability law).

[F3]

The expected resolvent has deterministic equivalent $- (z (I + q_{n} T_{n}))^{- 1}$ , uniformly for deterministic unit-vector quadratic forms (Deterministic equivalent for the bounded complex sample-covariance resolvent).

[F4]

The bounded-entry empirical spectral distribution converges almost surely to $F_{c, H}$ with the stated fixed-point transform (General-population sample-covariance law for bounded entries).

[F5]

A bounded-entry directional resolvent quadratic form concentrates almost surely about its expectation (Adapted concentration of bounded complex resolvent quadratic forms).

[F6]

Dominated convergence passes limits through integrals under a common integrable bound (Dominated convergence).

[F7]

The normalized operator norm of bounded complex iid matrices is almost surely eventually bounded (Operator-norm bound for bounded complex iid matrices).

[F8]

Tight probability laws on a Polish space have weakly convergent subsequences under AC (Prokhorov tightness theorem on polish spaces).

[F9]

The Stieltjes transform uses $(t - z)^{- 1}$ and determines a probability measure uniquely (Stieltjes transform of a probability measure on the real line, Stieltjes inversion and uniqueness for probability measures on the real line).

[F10]

The nonzero eigenvalues of $Y^{*} Y$ and $Y Y^{*}$ agree with multiplicity (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F11]

A Polish space is a separable topological space whose topology is induced by a complete metric (Polish spaces are separable completely metrizable spaces).

[F12]

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

[F13]

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

1.1givenF4F6F9F10algebra

Fix $z \in C^{+}$ and let $q_{n} (z) = E [N_{n}^{- 1} tr (N_{n}^{- 1} X_{n}^{*} T_{n} X_{n} - z I)^{- 1}]$ . By [F4], the empirical transforms of $A_{n}$ converge almost surely to $m_{c, H} (z)$ ; their absolute values are at most $1 / Im z$ , so [F6] gives convergence of their expectations. By [F10], the nonzero eigenvalue equality for $A_{n}$ and $N_{n}^{- 1} X_{n}^{*} T_{n} X_{n}$ gives $q_{n} = - (1 - c_{n}) / z + c_{n} E [n^{- 1} tr (A_{n} - z I)^{- 1}]$ , with $c_{n} = n / N_{n}$ . Hence $q_{n} (z) \to u (z) : = - (1 - c) / z + c m_{c, H} (z)$ .

2.1givenF3F5F13step 1.1algebra

For deterministic unit $x_{n}$ , [F3] and [F5] yield $x_{n}^{*} (A_{n} - z I)^{- 1} x_{n} + \frac{1}{z} x_{n}^{*} (I + q_{n} (z) T_{n})^{- 1} x_{n} ⟶ 0 almost surely .$ The matrix inverses are uniformly bounded by $∣ z ∣ / Im z$ from the positive-measure resolvent estimate in the proof of [F3]. Since $T_{n}$ are uniformly bounded and $q_{n} \to u$ , the resolvent identity gives ${∥ (I + q_{n} T_{n})^{- 1} - (I + u T_{n})^{- 1} ∥}_{o p} \leq M (∣ z ∣ / Im z)^{2} ∣ q_{n} - u ∣ ⟶ 0.$

3.1givenF4step 1.1step 2.1algebra

Since $u \in C^{+}$ , $w = - 1 / u$ also lies in $C^{+}$ and $(I + u T_{n})^{- 1} = u^{- 1} (T_{n} - w I)^{- 1}$ . The population-resolvent hypothesis at this fixed $w$ therefore gives $x_{n}^{*} (I + u T_{n})^{- 1} x_{n} \to u^{- 1} m_{H} (- 1 / u)$ . By [F4], $m_{c, H} (z) = - z^{- 1} \int (1 + u t)^{- 1} d H (t) = - [z u]^{- 1} m_{H} (- 1 / u)$ . Step 2.1 now proves $x_{n}^{*} (A_{n} - z I)^{- 1} x_{n} \to m_{c, H} (z)$ almost surely for each fixed $z \in C^{+}$ .

4.1givenA1F1F2F7F8F9F11F12F13step 3.1algebra∎

The resolvent in step 3.1 is the Stieltjes transform of $μ_{n}^{x_{n}}$ by [F1]. Intersect the probability-one events over a countable dense subset of $C^{+}$ . By [F7], on this event the spectra of $A_{n}$ are eventually contained in one compact interval, so [F2] makes the directional measures a tight sequence of probabilities. The half-line is Polish: it is a closed subset of the complete real line with dense subset $Q \cap [0, \infty)$ , in the sense of [F11]. By [A1] and [F8], every subsequence has a weakly convergent further subsequence. Its transform agrees with $m_{c, H}$ on the dense set by step 3.1; continuity of Stieltjes transforms extends equality to all $C^{+}$ , and [F9] identifies that limit with $F_{c, H}$ . Therefore every subsequence has the same further limit, which by the bounded-continuous-test definition [F12] proves $μ_{n}^{x_{n}} \Rightarrow F_{c, H}$ almost surely.

Source notes

Bai, Miao, and Pan, Theorem 1, equations (1.1)–(1.5), and Section 2, equations (2.1)–(2.9), printed pp. 8–16 (full paper PDF).

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

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.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Sample-covariance resolvent deterministic equivalents

Statement

Assume the Axiom of Choice. Let $(ξ_{i j})_{i, j \geq 1}$ be one infinite iid array of complex random variables with $E ξ_{11} = 0$ , $E ∣ ξ_{11} ∣^{2} = 1$ , and $E ∣ ξ_{11} ∣^{4} < \infty$ . For each positive integer $n$ , let $N_{n}$ be positive with $n / N_{n} \to c \in (0, \infty)$ and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq N_{n}}$ be its upper-left block. Let $T_{n}$ be deterministic Hermitian positive semidefinite with uniformly bounded operator norm and empirical eigenvalue measures $H_{n} = n^{- 1} \sum_{i} δ_{λ_{i} (T_{n})} \Rightarrow H$ . Let $x_{n} \in C^{n}$ be deterministic unit vectors such that, for every $w \in C ∖ [0, \infty)$ , $x_{n}^{*} (T_{n} - w I_{n})^{- 1} x_{n} ⟶ m_{H} (w) : = \int_{[0, \infty)} (t - w)^{- 1} d H (t) .$ Define $A_{n} = N_{n}^{- 1} T_{n}^{1 / 2} X_{n} X_{n}^{*} T_{n}^{1 / 2}$ and $μ_{n}^{x_{n}} (B) = x_{n}^{*} 1_{B} (A_{n}) x_{n}$ for Borel $B \subseteq [0, \infty)$ . Then almost surely $μ_{n}^{x_{n}}$ converges weakly to the probability measure $F_{c, H}$ whose Stieltjes transform is the unique $m \in C$ satisfying $Im m > 0$ , $u = - (1 - c) / z + c m$ with $Im u > 0$ , and $m = \int_{[0, \infty)} \frac{1}{t (1 - c - c z m) - z} d H (t), Im z > 0.$

Statement references

Bai, Miao, and Pan, Theorem 1 and complete Section 2 proof, printed pp. 8–16 (full paper PDF).

Silverstein, Theorem 1.1 and complete Section 3 proof (full paper PDF).

Facts & Assumptions

Given: The common nested iid array, finite-fourth-moment law, deterministic population matrices and directions, aspect ratio, and AC in the Statement.

[A1]

The Axiom of Choice permits a choice function for every family of nonempty sets (The Axiom of Choice).

[F1]

For bounded complex iid entries with the stated population and direction hypotheses, the directional measures converge almost surely to $F_{c, H}$ (Directional deterministic equivalent for bounded sample-covariance entries).

[F2]

For a centered complex iid finite-fourth-moment array with $E ∣ ξ_{11} ∣^{2} = τ^{2}$ , the same nested-block and aspect-ratio conventions give $lim sup ∥ X_{n} ∥_{o p} / \sqrt{N_{n}} \leq (σ_{R} + σ_{I}) (1 + \sqrt{c}) \leq \sqrt{2} τ (1 + \sqrt{c})$ almost surely, where $σ_{R}^{2} = E (Re ξ_{11})^{2}$ and $σ_{I}^{2} = E (Im ξ_{11})^{2}$ ; the bound includes $τ = 0$ (A coarse complex sample-covariance norm bound from real edge bounds).

[F3]

A Hermitian directional spectral measure is a probability law (A Hermitian directional spectral measure is a probability law).

[F4]

Tight probability laws on a Polish space have weakly convergent subsequences under AC (Prokhorov tightness theorem on polish spaces).

[F5]

Equal Stieltjes transforms determine probability measures on the real line (Stieltjes inversion and uniqueness for probability measures on the real line).

[F6]

A Polish space is a separable topological space whose topology is induced by a complete metric (Polish spaces are separable completely metrizable spaces).

[F7]

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

[F8]

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

[F9]

The Stieltjes transform convention is $m_{μ} (z) = \int (t - z)^{- 1} d μ (t)$ on $C^{+}$ (Stieltjes transform of a probability measure on the real line).

[F10]

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

[F11]

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

1.1givenF1F2F7algebra

Let $ξ$ have the common entry law. For integer $K$ , set $ξ^{(K)} = ξ 1_{{∣ ξ ∣ \leq K}} - E [ξ 1_{{∣ ξ ∣ \leq K}}]$ , $σ_{K}^{2} = E ∣ ξ^{(K)} ∣^{2}$ , and $Δ^{(K)} = ξ - ξ^{(K)}$ . Then $σ_{K} \to 1$ , and $τ_{K}^{2} : = E ∣ Δ^{(K)} ∣^{2} \to 0$ because $E ξ = 0$ and $E ∣ ξ ∣^{2} < \infty$ . For all sufficiently large $K$ , $σ_{K} > 0$ ; put $ζ^{(K)} = ξ^{(K)} / σ_{K}$ . These entries are centered, bounded, iid and have unit second absolute moment, while $Δ^{(K)}$ is centered iid with finite fourth moment. At matrix level, $X_{n} = σ_{K} Z_{n}^{(K)} + D_{n}^{(K)}$ , where $Z_{n}^{(K)}$ and $D_{n}^{(K)}$ are the common upper-left blocks formed from $ζ^{(K)}$ and $Δ^{(K)}$ .

2.1givenF2F11step 1.1algebra

For each fixed sufficiently large integer $K$ , the original, standardized, and residual arrays have second absolute moments $1$ , $1$ , and $τ_{K}^{2}$ , respectively. Applying the variance-scaled bound [F2] to each gives almost surely ${lim sup}_{n} \frac{∥ X_{n} ∥}{\sqrt{N_{n}}} \leq C_{c}, {lim sup}_{n} \frac{∥ Z_{n}^{(K)} ∥}{\sqrt{N_{n}}} \leq C_{c}, {lim sup}_{n} \frac{∥ D_{n}^{(K)} ∥}{\sqrt{N_{n}}} \leq C_{c} τ_{K},$ where $C_{c} = \sqrt{2} (1 + \sqrt{c})$ . If $τ_{K} = 0$ , the centered residual entries are zero almost surely; countability of the double array gives $D_{n}^{(K)} = 0$ for every $n$ on one probability-one event, so the residual bound still holds. Intersect these events over the countable set of such $K$ . Writing $A_{n}^{(K)} = N_{n}^{- 1} T_{n}^{1 / 2} Z_{n}^{(K)} Z_{n}^{(K) *} T_{n}^{1 / 2}$ , the identity $U U^{*} - V V^{*} = (U - V) U^{*} + V (U - V)^{*}$ yields ${lim sup}_{n} ∥ A_{n} - σ_{K}^{2} A_{n}^{(K)} ∥_{o p} \leq M C_{c}^{2} (1 + σ_{K}) τ_{K}, {lim sup}_{n} ∥ σ_{K}^{2} A_{n}^{(K)} - A_{n}^{(K)} ∥_{o p} \leq M C_{c}^{2} ∣ σ_{K}^{2} - 1 ∣,$ where $M = \sup_{n} ∥ T_{n} ∥_{o p}$ . For positive semidefinite $B, C$ and $z \in C^{+}$ , the resolvent identity gives $∥ (B - z I)^{- 1} - (C - z I)^{- 1} ∥_{o p} \leq ∥ B - C ∥_{o p} / (Im z)^{2}$ .

3.1givenF1F9step 1.1step 2.1algebra

Fix $z \in C^{+}$ and a deterministic unit direction sequence $x_{n}$ . For each such $K$ , [F1] applied to $Z_{n}^{(K)}$ gives $x_{n}^{*} (A_{n}^{(K)} - z I)^{- 1} x_{n} \to m_{c, H} (z)$ almost surely, with the transform convention [F9]. By step 2.1, ${lim sup}_{n} ∣ x_{n}^{*} (A_{n} - z I)^{- 1} x_{n} - m_{c, H} (z) ∣ \leq \frac{M C_{c}^{2}}{(Im z)^{2}} ((1 + σ_{K}) τ_{K} + ∣ σ_{K}^{2} - 1 ∣)$ on the probability-one event obtained by intersecting the events from step 2.1 with the almost-sure convergence event for this fixed truncation level. The right side tends to zero as $K \to \infty$ , so the directional Stieltjes transforms converge almost surely at every fixed $z \in C^{+}$ . Intersecting over a countable dense set gives simultaneous convergence there.

4.1givenA1F2F3F4F5F6F8F10F11step 3.1algebra∎

By [F2], $∥ A_{n} ∥_{o p} \leq M ∥ X_{n} ∥_{o p}^{2} / N_{n}$ is almost surely eventually bounded. Thus the directional laws are eventually supported on one compact interval; [F3] makes them probability measures and [F8] gives their exact spectral-measure meaning. They are therefore tight. The half-line is Polish: it is closed in the complete real line and has dense subset $Q \cap [0, \infty)$ , as required by [F6]. By [A1] and [F4], each subsequence has a weakly convergent further subsequence. Its transform equals $m_{c, H}$ on the dense set by step 3.1; continuity of Stieltjes transforms extends equality to $C^{+}$ , and [F5] identifies the subsequential limit as $F_{c, H}$ . The bounded-continuous-test definition [F10] yields the stated weak convergence. Every subsequence has the same further limit, proving the asserted almost-sure limit.

Source notes

Bai, Miao, and Pan, Theorem 1 and Section 2, printed pp. 8–16 (full paper PDF). The sampling convention is the upper-left block of one iid double array. The truncation comparison uses the separately proved real-edge norm supplier; it does not assume weak convergence alone controls spectral edges.

CorollaryStatement: AI-adaptedProof: AI-generatedOpen item page →

Isotropic minimum-norm risk limits under a positive spectral gap

Statement

Assume the Axiom of Choice. 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 $n$ , let $p_{n}$ be a positive integer with $p_{n} / n \to γ \in (0, \infty) ∖ {1}$ and let $X_{n} = (ξ_{i j})_{1 \leq i \leq n, 1 \leq j \leq p_{n}}$ be the upper-left block. Let $β_{n} \in R^{p_{n}}$ be deterministic with $∥ β_{n} ∥_{2} = r > 0$ . Let $ε_{n} \in R^{n}$ have independent centered components of variance $σ^{2}$ , independent of $X_{n}$ (so its covariance is $σ^{2} I_{n}$ ), and let $z_{0, n} \in R^{p_{n}}$ be independent of $(X_{n}, ε_{n})$ with mean zero and covariance $I_{p_{n}}$ . Set $y_{n} = X_{n} β_{n} + ε_{n}$ , ${\hat{β}}_{n} = X_{n}^{+} y_{n}$ , and $S_{n} = X_{n}^{T} X_{n} / n$ . Define the design-conditional prediction risk by $R_{n} = E_{ε_{n}, z_{0, n}} [(z_{0, n}^{T} {\hat{β}}_{n} - z_{0, n}^{T} β_{n})^{2} ∣ X_{n}]$ . If $γ < 1$ , then almost surely $R_{n} \to σ^{2} γ / (1 - γ)$ . If $γ > 1$ , then almost surely $R_{n} \to r^{2} (1 - 1 / γ) + σ^{2} / (γ - 1)$ .

Statement references

Hastie, Montanari, Rosset, and Tibshirani, Theorem 1 and Appendix C.2 (full paper PDF).

Bai, Miao, and Pan, Theorem 1 (full paper PDF).

Facts & Assumptions

Given: The nested real iid design, deterministic signal, independent centered noise with covariance $σ^{2} I_{n}$ , and AC in the Statement.

[A1]

The Axiom of Choice permits a choice function for every family of nonempty sets (The Axiom of Choice).

[A2]

The noise vector is independent of the design, centered, and has covariance $σ^{2} I_{n}$ as stated.

[A3]

The test feature is independent of the design and noise, centered, and has covariance $I_{p_{n}}$ as stated.

[F1]

The Moore–Penrose inverse $X^{+}$ is characterized by the four Penrose equations (The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations).

[F2]

$X^{+} X$ and $X X^{+}$ are orthogonal projections with ranges $Ran (X^{+})$ and $Ran (X)$ , respectively (The Moore-Penrose products are orthogonal projectors).

[F3]

Positive covariance eigenvalues are squared positive singular values divided by the sample size, and their number is the matrix rank (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition, The rank of a linear map is the number of its nonzero singular values).

[F4]

For the nested design, the smallest positive covariance eigenvalue has limit $(1 - \sqrt{γ})^{2}$ for $γ < 1$ and $(\sqrt{γ} - 1)^{2}$ for $γ > 1$ , with eventual full rectangular rank (Almost-sure limit and structural-zero count at the lower sample-covariance edge).

[F5]

The empirical spectral distribution of $S_{n}$ converges almost surely to the Marchenko–Pastur law with aspect ratio $γ$ (Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices).

[F6]

For upper-left $d \times N_{d}$ blocks indexed by their row dimension $d$ , the directional theorem with identity population gives almost-sure weak convergence to $F_{γ, δ_{1}}$ when $d / N_{d} \to γ$ (Sample-covariance resolvent deterministic equivalents). Repeated dimensions $p_{n}$ require the argument below.

[F7]

The Marchenko–Pastur probability law has an atom $(1 - 1 / γ)_{+}$ at zero, positive support $[(1 - \sqrt{γ})^{2}, (1 + \sqrt{γ})^{2}]$ , and transform satisfying $1 + z m = m / (1 + γ m)$ (Marchenko–Pastur measure with aspect ratio gamma, The Marchenko–Pastur formula defines a probability law with the fixed-point transform).

[F8]

Weak convergence gives convergence of integrals against every bounded continuous test function (Weak convergence of borel probability measures).

[F9]

The deformed fixed-point solution is unique when both its transform and its companion transform have positive imaginary part (Uniqueness of the deformed Marchenko–Pastur fixed point).

[F10]

Dominated convergence passes limits through integrals under an integrable majorant (Dominated convergence).

[F11]

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

[F12]

The finite-column proof of bounded resolvent concentration, steps 1.1–4.1, bounds the fourth moment of each leave-one-column increment by $C / N^{4}$ and that of the centered directional resolvent by $C / N^{2}$ , with constants independent of the unit direction and row dimension when the population is the identity (Adapted concentration of bounded complex resolvent quadratic forms).

[F13]

For a nested real iid unit-variance finite-fourth-moment array with $p_{n} / n \to γ$ , almost surely $lim sup ∥ X_{n} ∥_{o p} / \sqrt{n} \leq 1 + \sqrt{γ}$ (Finite-fourth-moment upper bound for the largest sample-covariance eigenvalue, [F3]).

[F14]

Markov bounds a nonnegative random variable's tail by its expectation divided by the threshold; summable event probabilities imply only finitely many occurrences almost surely (Markov's inequality for random variables, First Borel-Cantelli lemma for events).

Proof

1.1givenA2A3F1F2F3algebra

Put $P_{\ker X_{n}} = I_{p_{n}} - X_{n}^{+} X_{n}$ . For any fixed $v \in R^{p_{n}}$ , $E_{z_{0, n}} (z_{0, n}^{T} v)^{2} = v^{T} E [z_{0, n} z_{0, n}^{T}] v = ∥ v ∥_{2}^{2}$ , so the stated prediction risk equals $E_{ε_{n}} [∥ {\hat{β}}_{n} - β_{n} ∥_{2}^{2} ∣ X_{n}]$ . By [F2], $P_{n} : = X_{n}^{+} X_{n}$ is an orthogonal projection. The Penrose equation $X_{n} X_{n}^{+} X_{n} = X_{n}$ shows $\ker (P_{n}) = \ker (X_{n})$ : if $X_{n} v = 0$ , then $P_{n} v = 0$ , and if $P_{n} v = 0$ , then $X_{n} v = X_{n} X_{n}^{+} X_{n} v = X_{n} P_{n} v = 0$ . Hence $Ran (P_{n}) = \ker (X_{n})^{⊥} = Ran (X_{n}^{T})$ by the right-singular-vector basis in [F3], so ${\hat{β}}_{n} - β_{n} = - P_{\ker X_{n}} β_{n} + X_{n}^{+} ε_{n}$ . Conditional on $X_{n}$ , the cross term has expectation zero because $E [ε_{n} ∣ X_{n}] = 0$ . Its noise contribution is $E_{ε_{n}} ∥ X_{n}^{+} ε_{n} ∥_{2}^{2} = σ^{2} tr (X_{n}^{+} (X_{n}^{+})^{T}) = σ^{2} tr ((X_{n}^{T} X_{n})^{+}) = \frac{σ^{2}}{n} tr (S_{n}^{+}),$ where the middle equality follows by writing an SVD and the last uses $(a B)^{+} = a^{- 1} B^{+}$ for $a > 0$ . With $x_{n} = β_{n} / r$ , this proves the exact identity $R_{n} = r^{2} x_{n}^{T} P_{\ker X_{n}} x_{n} + \frac{σ^{2}}{n} tr (S_{n}^{+}) .$

1.2givenA1F6F7F9F10F11algebra

First identify the isotropic limit. By [F7], $ν_{γ} = (1 - γ) δ_{0} + γ μ_{γ}$ is a nonnegative probability measure: when $γ > 1$ , the negative coefficient cancels exactly the zero atom of $γ μ_{γ}$ . Thus $u = - (1 - γ) / z + γ m_{γ} (z)$ is its Stieltjes transform and has positive imaginary part, as does $m_{γ} (z)$ . The scalar equation in [F7] rearranges to $m_{γ} = (1 - γ - z - γ z m_{γ})^{- 1}$ ; [F9] therefore identifies $F_{γ, δ_{1}} = μ_{γ}$ . For any deterministic unit directions $x_{n}$ and fixed $z \in C^{+}$ , we claim $E [x_{n}^{T} (S_{n} - z I)^{- 1} x_{n}] \to m_{γ} (z)$ . Indeed, every subsequence has a further subsequence $(n_{j})$ with strictly increasing $p_{n_{j}}$ , since $p_{n} \to \infty$ . In the transposed infinite array, prescribe $N_{d} = n_{j}$ and direction $x_{n_{j}}$ at $d = p_{n_{j}}$ , and at all other dimensions prescribe $N_{d} = \max (1, ⌊ d / γ ⌋)$ and the first coordinate vector. Then $d / N_{d} \to γ$ , so [F6] applies to this completed sequence. The bound $∣ (t - z)^{- 1} ∣ \leq 1 / Im z$ and [F10] give the claimed expectation limit along $(n_{j})$ . The subsequence criterion gives it along the original sequence. This argument also applies to each fixed standardized bounded truncation of the entry law.

2.1givenA1F4F5F7F8F10F11step 1.1algebra

Suppose $γ < 1$ and set $a = (1 - \sqrt{γ})^{2} > 0$ . By [F4], $X_{n}$ has full column rank eventually and every eigenvalue of $S_{n}$ is at least $a / 2$ eventually. Thus the bias term in step 1.1 is zero and $p_{n}^{- 1} tr (S_{n}^{+}) = \int f (t) d μ_{n} (t)$ for the bounded continuous function $f (t) = 1 / \max (t, a / 2)$ . By [F5] and [F8], this converges to $J_{γ} = \int t^{- 1} d μ_{γ} (t)$ , since the Marchenko–Pastur support is bounded below by $a$ . For $s > 0$ , define $m_{γ} (- s) = \int (t + s)^{- 1} d μ_{γ} (t)$ . The integrands are bounded by $1 / a$ and converge pointwise to $1 / t$ , so [F10] gives $m_{γ} (- s) \to J_{γ}$ as $s ↓ 0$ . The fixed-point equation in [F7], initially stated for $z \in C^{+}$ , extends to $z = - s < 0$ by dominated convergence in its defining integral; its denominator stays nonzero because $1 + γ m_{γ} (- s) > 0$ . Taking $s ↓ 0$ yields $1 + (γ - 1) J_{γ} = 0$ , hence $J_{γ} = 1 / (1 - γ)$ . Since $p_{n} / n \to γ$ , step 1.1 gives $R_{n} \to σ^{2} γ / (1 - γ)$ .

2.2givenF12F14step 1.2algebra

For such a bounded unit-variance truncation, let $Z_{n}$ be its $n \times p_{n}$ block and $S_{n}^{(K)} = Z_{n}^{T} Z_{n} / n$ . Apply the finite-column argument [F12] to $Z_{n}^{T}$ , which has $n$ independent columns and identity population. More explicitly, each leave-one-column resolvent quadratic-form increment $U_{j}$ has $E ∣ U_{j} ∣^{4} \leq C_{K} n^{- 4}$ by the bounded-entry eighth-moment linear-form estimate; its column-filtration difference $D_{j}$ satisfies $E ∣ D_{j} ∣^{4} \leq 16 E ∣ U_{j} ∣^{4}$ . The fourth-moment recursion in [F12], $a_{j} \leq a_{j - 1} + C_{K} \sqrt{a_{j - 1}} / n^{2} + C_{K} / n^{4}$ , gives $a_{j} \leq C_{K}^{'} j^{2} / n^{4}$ , and hence $E ∣ x_{n}^{T} ((S_{n}^{(K)} - z I)^{- 1} - E (S_{n}^{(K)} - z I)^{- 1}) x_{n} ∣^{4} \leq C_{K}^{'} / n^{2}$ . These are finite-matrix estimates, with no indexing or distinctness assumption on $p_{n}$ . By [F14], their tails are summable in the original index $n$ . Together with step 1.2 this proves $x_{n}^{T} (S_{n}^{(K)} - z I)^{- 1} x_{n} \to m_{γ} (z)$ almost surely for each fixed $K, z$ .

3.1givenF3F11F13step 2.2algebra

To remove boundedness, put $ξ^{(K)} = ξ 1_{{∣ ξ ∣ \leq K}} - E [ξ 1_{{∣ ξ ∣ \leq K}}]$ , $σ_{K}^{2} = E ∣ ξ^{(K)} ∣^{2} \to 1$ , and $τ_{K}^{2} = E ∣ ξ - ξ^{(K)} ∣^{2} \to 0$ . For sufficiently large integers $K$ , form $Z_{n}$ from $ξ^{(K)} / σ_{K}$ and write $X_{n} = σ_{K} Z_{n} + D_{n}$ . The residual entries are centered with finite fourth moment and variance $τ_{K}^{2}$ . Applying [F13] to the original, standardized and variance-normalized residual arrays yields, with $L = 1 + \sqrt{γ}$ , almost surely $lim sup ∥ X_{n} ∥ / \sqrt{n} \leq L$ , $lim sup ∥ Z_{n} ∥ / \sqrt{n} \leq L$ , and $lim sup ∥ D_{n} ∥ / \sqrt{n} \leq L τ_{K}$ . If $τ_{K} = 0$ , countability of the array makes every residual entry zero almost surely. The Gram-matrix identity then gives $lim sup ∥ S_{n} - S_{n}^{(K)} ∥ \leq L^{2} ((1 + σ_{K}) τ_{K} + ∣ σ_{K}^{2} - 1 ∣)$ . For $η = Im z > 0$ , the resolvent identity bounds the difference of their directional transforms by $η^{- 2} ∥ S_{n} - S_{n}^{(K)} ∥$ . Intersect the probability-one events over integers $K$ and let $K \to \infty$ in step 2.2 to obtain $x_{n}^{T} (S_{n} - z I)^{- 1} x_{n} \to m_{γ} (z)$ almost surely.

4.1givenF3F4F5F7F8F10step 1.1step 3.1algebra∎

Suppose $γ > 1$ and set $a = (\sqrt{γ} - 1)^{2} > 0$ . By [F4], $X_{n}$ has full row rank eventually, $S_{n}$ has exactly $p_{n} - n$ zero eigenvalues, and all positive eigenvalues are at least $a / 2$ eventually. For each rational $s > 0$ , the spectral identity bounds the difference between the directional transforms at $- s$ and $- s + i η$ by $η / s^{2}$ , both for $S_{n}$ and for $μ_{γ}$ . Use step 3.1 at $- s + i / k$ for all positive integers $k$ , then let $k \to \infty$ , to obtain $x_{n}^{T} (S_{n} + s I)^{- 1} x_{n} \to m_{γ} (- s)$ on one probability-one event for all rational $s > 0$ . The gap gives $0 \leq s x_{n}^{T} (S_{n} + s I)^{- 1} x_{n} - x_{n}^{T} P_{\ker X_{n}} x_{n} \leq 2 s / a$ eventually. Since $s m_{γ} (- s) \to μ_{γ} ({0}) = 1 - 1 / γ$ by [F7] and [F10], letting rational $s ↓ 0$ proves the required bias limit. Define $h (0) = 0$ and, for $t > 0$ , $h (t) = t^{- 1} \min (1, \max (0, 4 t / a - 1))$ . This is bounded and continuous, equals zero at zero, and equals $1 / t$ for $t \geq a / 2$ . Therefore $p_{n}^{- 1} tr (S_{n}^{+}) = \int h (t) d μ_{n} (t) \to J_{γ}^{+} : = \int_{(0, \infty)} t^{- 1} d μ_{γ} (t)$ by [F5] and [F8]. [F7] gives $m_{γ} (- s) = (1 - 1 / γ) / s + J_{γ}^{+} + o (1)$ as $s ↓ 0$ : subtract the zero atom and use [F10] by dominated convergence on the positive support, whose lower endpoint is $a > 0$ . To compute the constant term, write $α = 1 - 1 / γ$ and substitute $m = α / s + J_{γ}^{+} + o (1)$ into $1 + (γ - 1 - s) m - γ s m^{2} = 0$ , the fixed-point equation at $z = - s$ (again obtained by dominated convergence from [F7]). The coefficient of $s^{- 1}$ vanishes because $γ α = γ - 1$ ; the constant term is $1 - α - (γ - 1) J_{γ}^{+} + o (1)$ . It must vanish, so $J_{γ}^{+} = 1 / [γ (γ - 1)]$ . Using $p_{n} / n \to γ$ in step 1.1 proves $R_{n} \to r^{2} (1 - 1 / γ) + σ^{2} / (γ - 1)$ .

Source notes

Hastie, Montanari, Rosset, and Tibshirani, Theorem 1 and Appendix C.2 (full paper PDF). The repeated-dimension argument in steps 1.2, 2.2, and 3.1 uses injective row dimensions only for expectation limits; almost-sure concentration is summed in the original sample index. It follows the truncation and martingale structure of Bai, Miao, and Pan, Section 2, without changing the row-indexed supplier interface. The proof above separates the conditional noise expectation from the design almost-sure limits and uses a positive-edge cutoff rather than a $z ↓ 0$ interchange.

CorollaryStatement: AI-adaptedProof: AI-generatedjudge pass (gpt-6-luna)Open item page →

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.

4 · Examples, counterexamples and false statements

None yet.