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

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Basic finite covariance matrix identities for robust mean bounds

Statement

Let $d \geq 1$ and let $X$ be a Borel random vector in $R^{d}$ with $E ∥ X ∥_{2}^{2} < \infty$ . Define $μ = E X$ coordinatewise, $Y = X - μ$ , and the finite matrix $Σ$ by $Σ_{r s} = E [Y_{r} Y_{s}]$ for $1 \leq r, s \leq d$ . Then $Σ$ is symmetric positive semidefinite,

tr Σ = E ∥ X - μ ∥_{2}^{2},

and $0 \leq λ_{\max} (Σ) \leq tr Σ$ . If $tr Σ = 0$ , then $X = μ$ almost surely.

Facts & Assumptions

Given: A Borel random vector $X$ in $R^{d}$ , $d \geq 1$ , with finite second moment.

[F1]

The Borel random-vector hypothesis means inverse images of Borel sets are measurable; composing with the continuous coordinate and norm-square maps preserves this property because inverse images of open sets under continuous maps are open. Expectation is integration against the probability measure and variance is the second centered moment (Random elements and real random variables, Expectation of a nonnegative or integrable random variable, Moments, variance, and covariance on a probability space).

[F2]

Expectation is linear for integrable variables (Linearity, monotonicity, and the modulus bound for expectation). The pointwise inequalities $∣ x ∣ \leq 1 + x^{2}$ and $2 ∣ x y ∣ \leq x^{2} + y^{2}$ give the integrability bounds used below.

[F3]

The Euclidean inner product is $⟨ x, y ⟩ = \sum_{r = 1}^{d} x_{r} y_{r}$ and $∥ x ∥_{2}^{2} = ⟨ x, x ⟩$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

A nonnegative measurable function with integral zero vanishes almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F5]

Cauchy–Schwarz holds in the Euclidean inner product, and a symmetric matrix in the standard orthonormal basis represents a self-adjoint endomorphism whose largest eigenvalue is the maximum of its Rayleigh quotient $R_{Σ} (v) = ⟨ Σ v, v ⟩ / ⟨ v, v ⟩$ over nonzero $v$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, The Rayleigh quotient of a nonzero vector for a self-adjoint endomorphism, The smallest and largest eigenvalues of a self-adjoint endomorphism are the minimum and maximum Rayleigh quotients).

Proof

technique · direct

Given: A Borel random vector $X$ in $R^{d}$ , $d \geq 1$ , with $E ∥ X ∥_{2}^{2} < \infty$ .

1.1F1F3givenalgebra

Each coordinate $X_{r}$ is square-integrable because $∣ X_{r} ∣ \leq ∥ X ∥_{2}$ , and is integrable since $∣ X_{r} ∣ \leq 1 + X_{r}^{2}$ . Hence each $μ_{r}$ is finite. Also $E [Y_{r}^{2}] \leq 2 E [X_{r}^{2}] + 2 μ_{r}^{2} < \infty$ . For every $r, s$ , the pointwise inequality $2 ∣ Y_{r} Y_{s} ∣ \leq Y_{r}^{2} + Y_{s}^{2}$ gives $E ∣ Y_{r} Y_{s} ∣ < \infty$ , so every entry of $Σ$ exists. Since $Y_{r} Y_{s} = Y_{s} Y_{r}$ , $Σ$ is symmetric.

2.1F2F3step 1.1algebra

For $v \in R^{d}$ , linearity of expectation [F2] and the finite-sum definition [F3] give $v^{T} Σ v = E [⟨ v, Y ⟩^{2}] \geq 0$ , so $Σ$ is positive semidefinite. Summing its diagonal entries and using linearity again yields $tr Σ = E [\sum_{r = 1}^{d} Y_{r}^{2}] = E ∥ Y ∥_{2}^{2}$ .

3.1F3F5step 2.1algebra

For nonzero $v$ , [F5] identifies $λ_{\max} (Σ)$ with the maximum Rayleigh quotient; pointwise Euclidean Cauchy–Schwarz gives $R_{Σ} (v) = E [⟨ v, Y ⟩^{2}] / ∥ v ∥_{2}^{2} \leq E ∥ Y ∥_{2}^{2} = tr Σ$ . Positive semidefiniteness from step 2.1 also gives $R_{Σ} (v) \geq 0$ , so maximizing proves $0 \leq λ_{\max} (Σ) \leq tr Σ$ .

4.1F1F4step 2.1∎

If $tr Σ = 0$ , step 2.1 gives $E ∥ Y ∥_{2}^{2} = 0$ . By [F1], the continuous polynomial $x \mapsto ∥ x - μ ∥_{2}^{2}$ composed with the Borel map $X$ is measurable, so [F4] applies and gives $∥ Y ∥_{2}^{2} = 0$ almost surely. The Euclidean norm is positive definite, hence $Y = 0$ almost surely and $X = μ$ almost surely.

Depends on

Used by

The four-Q0 trimming slab radius is false Counterexample
Exact recovery by the multivariate trimmed mean at zero covariance Lemma
The multivariate trimmed-mean rate follows from a correct-scale slab event Lemma
Correct-scale slab bound for multivariate trimming Proposition
Multivariate trimmed-mean bound under sample replacement Theorem

Dependency tree · two levels

49 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

Stanislav Minsker, Geometric Median and Robust Estimation in Banach Spaces, Corollary 4.1 (standard reference, not scraped)
Gábor Lugosi and Shahar Mendelson, Robust Multivariate Mean Estimation: The Optimality of Trimmed Mean, Theorem 2 (standard reference, not scraped)