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

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A finite second moment defines a positive semidefinite covariance matrix

Statement

If $X \in R^{p}$ is measurable and $E ∥ X ∥_{2}^{2} < \infty$ , then $μ = E X$ and $Σ = E [(X - μ) (X - μ)^{T}]$ in Covariance matrix of a finite-dimensional random vector are well-defined entrywise, $Σ$ is symmetric positive semidefinite, and $tr (Σ) = E ∥ X - μ ∥_{2}^{2}$ .

Facts & Assumptions

Given: A finite-dimensional measurable random vector $X$ with $E ∥ X ∥_{2}^{2} < \infty$ .

[F1]

Covariance matrix of a finite-dimensional random vector defines the mean coordinatewise and covariance entries by $E [(X_{j} - μ_{j}) (X_{k} - μ_{k})]$ .

[F2]

Expectation of a nonnegative or integrable random variable: finite expectation is defined for integrable real random variables.

Proof

1.1givenF1F2algebra

For each coordinate, $∣ X_{j} ∣ \leq ∥ X ∥_{2}$ and $∣ x ∣ \leq (1 + x^{2}) / 2$ imply $E ∣ X_{j} ∣ < \infty$ , so every $μ_{j}$ exists. Also $E (X_{j} - μ_{j})^{2} < \infty$ , since $(X_{j} - μ_{j})^{2} \leq 2 X_{j}^{2} + 2 μ_{j}^{2}$ .

2.1step 1.1F1F2algebra

For every pair $j, k$ , $2 ∣ (X_{j} - μ_{j}) (X_{k} - μ_{k}) ∣ \leq (X_{j} - μ_{j})^{2} + (X_{k} - μ_{k})^{2}$ . Thus every covariance entry is integrable; all entries are finite and symmetry follows from commutativity of multiplication.

3.1step 2.1F1F2algebra∎

For $a \in R^{p}$ , $a^{T} Σ a = E [(a^{T} (X - μ))^{2}] \geq 0$ , so $Σ$ is positive semidefinite. Summing the diagonal entries gives $tr (Σ) = \sum_{j} E (X_{j} - μ_{j})^{2} = E \sum_{j} (X_{j} - μ_{j})^{2} = E ∥ X - μ ∥_{2}^{2}$ ; the sum is finite.

Depends on

Used by

Cited to discharge well-definedness by Covariance matrix of a finite-dimensional random vector.

Dependency tree · two levels

8 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

Philippe Rigollet, 18.S997 High-Dimensional Statistics Lecture Notes, MIT OpenCourseWare, Chapter 4 (standard reference, not scraped)