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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).

Depends on

Used by

General-population sample-covariance law for bounded entries 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)
Z. D. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. (standard reference, not scraped)