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

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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.

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Covariance kernels of square-integrable fields are positive semidefinite

Statement

Let $X = (X (s))_{s \in T}$ be a real random field on one probability space, with $X (s) \in L^{2}$ for every $s \in T$ , and put $K (s, t) : = Cov (X (s), X (t))$ . Then $K$ is symmetric and positive semidefinite: for every $n \geq 1$ , sites $s_{1}, \dots, s_{n} \in T$ , and real $a_{1}, \dots, a_{n}$ , $\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i} a_{j} K (s_{i}, s_{j}) \geq 0.$

Facts & Assumptions

Given: A real random field on one probability space whose coordinate variables are all square-integrable, a finite list of sites, and real coefficients.

[F1]

Random fields, stationarity, covariance, and variograms: each field coordinate is a real random variable on the common probability space.

[F2]

Moments, variance, and covariance on a probability space: for square-integrable real variables, variance is the expectation of the centered square and covariance is the expectation of the centered product.

[F3]

Variance and covariance identities for random variables: finite linear combinations of square-integrable real variables remain square-integrable, and covariance is symmetric and bilinear on such finite linear combinations.

Proof

Given: The field and finite data in the statement.

1.1givenF1F2algebra

Put $W : = \sum_{i = 1}^{n} a_{i} X (s_{i})$ . The pointwise inequality $W^{2} \leq n \sum_{i = 1}^{n} a_{i}^{2} X (s_{i})^{2}$ shows that $W \in L^{2}$ , since each of the finitely many terms on the right is integrable by the given square-integrability. Thus $Var (W)$ is defined by [F2].

2.1F2F3step 1.1∎

Covariance bilinearity in [F3] gives $Var (W) = Cov (W, W) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i} a_{j} Cov (X (s_{i}), X (s_{j})) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i} a_{j} K (s_{i}, s_{j}) .$ The variance is nonnegative because its defining integrand is a square. The same covariance symmetry in [F3] gives $K (s, t) = K (t, s)$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Evgeny Spodarev, Spatial Statistics: Lecture Notes (2024) (standard reference, not scraped)
Hermine Biermé, Introduction to random fields and scale invariance: Lecture I (2016) (standard reference, not scraped)