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

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.

Adapted resolvent-trace martingale concentration for independent columns

Statement

Assume the Axiom of Choice, and let all random vectors below be defined on one common probability space. For each positive integer $n$ , let $d_{n}$ be a positive integer, let $Y_{n, 1}, \dots, Y_{n, n}$ be independent random vectors in $R^{d_{n}}$ , let $B_{n} = \sum_{j = 1}^{n} Y_{n, j} Y_{n, j}^{T}$ , and let $z = u + i η$ with fixed $u \in R$ and $η > 0$ . Define $M_{n} (z) : = \frac{1}{d_{n}} tr (B_{n} - z I)^{- 1} .$ If $d_{n} / n \to c$ for some $c \in (0, \infty)$ , then $M_{n} (z) - E [M_{n} (z)] \to 0$ almost surely. More precisely, for every $t > 0$ , $\Pr {∣ M_{n} (z) - E M_{n} (z) ∣ > t} \leq 4 \exp (- \frac{t^{2} d_{n}^{2} η^{2}}{4 n}) .$

Statement references

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

Junyi Liao, Chapter 2, §2.1.3, printed pp. 35–38 (complete lecture notes).

Facts & Assumptions

Given: AC, the common probability space, the positive integers $d_{n}$ , the independent column vectors $Y_{n, j}$ , and the fixed parameter $z$ with $Im z = η > 0$ .

[A1]

The Axiom of Choice states every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

For a real symmetric non-negative matrix, adding one rank-one matrix $y y^{T}$ changes the unnormalized resolvent trace by at most $1 / η$ (Rank-one resolvent update and trace bound for a positive semidefinite matrix).

[F2]

For independent random elements and a real measurable function whose change under replacement of coordinate $j$ is at most $c_{j}$ , the two-sided McDiarmid bound is $2 \exp (- 2 t^{2} / \sum_{j} c_{j}^{2})$ (McDiarmid bounded-differences inequality).

[F3]

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

Proof

Given: The hypotheses and notation of the Statement, including AC and the common probability space.

1.1givenF1algebra

Fix all vectors except $Y_{n, j}$ . Replacing that vector by another vector $Y_{n, j}^{'}$ changes $B_{n}$ by removing $Y_{n, j} Y_{n, j}^{T}$ and adding $Y_{n, j}^{'} (Y_{n, j}^{'})^{T}$ . Applying [F1] once to remove the first term and once to add the second shows that $tr (B_{n} - z I)^{- 1}$ changes by at most $2 / η$ . Hence each of the real and imaginary parts of $M_{n} (z)$ has coordinate oscillation at most $c_{j} = 2 / (d_{n} η)$ .

2.1A1F2step 1.1algebra

Apply [F2] separately to $Re M_{n} (z)$ and $Im M_{n} (z)$ . If a complex number has modulus greater than $t$ , at least one of its real or imaginary parts has magnitude greater than $t / \sqrt{2}$ . The union bound therefore gives $\Pr {∣ M_{n} (z) - E M_{n} (z) ∣ > t} \leq 4 \exp (- \frac{t^{2} d_{n}^{2} η^{2}}{4 n}) .$ This is the stated estimate.

3.1givenstep 2.1algebra

Since $d_{n} / n \to c > 0$ , for each fixed $t > 0$ the exponent in step 2.1 is at most $- c_{t} n$ for all sufficiently large $n$ , for some $c_{t} > 0$ . The probabilities are therefore summable.

4.1F3step 3.1choose∎

Apply [F3] to the events in step 3.1 for each $t = 1 / k$ , $k \geq 1$ , and intersect the resulting countable collection of probability-one events. On that event $M_{n} (z) - E M_{n} (z) \to 0$ .

Depends on

Used by

Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices Theorem

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Chapter 3, §3.3 (standard reference, not scraped)
Junyi Liao, Random Matrix Theory, Chapter 2, §2.1.3 (standard reference, not scraped)