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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).

Depends on

Used by

Sample-covariance resolvent deterministic equivalents Theorem

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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)