LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Characteristic functions are positive definite

Statement

Every characteristic function is positive definite.

Facts & Assumptions

Given: The hypotheses and conventions in the statement.

[F1]

The characteristic function integrates the exponential. Characteristic function of a real random variable.

[F2]

Positive definiteness is the finite nonnegative quadratic-form condition. Positive definite function on the real line.

[F3]

Products of exponentials add exponents. $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential.

[F4]

Integration commutes with finite complex linear combinations. The Lebesgue integral is linear on $L^{1} (μ)$ .

Proof

technique · direct

1.1

Fix $n \geq 1$ , real $t_{j}$ and complex $z_{j}$ . Set $Z (x) = \sum_{j = 1}^{n} z_{j} e^{i t_{j} x}$ . By the unit-modulus formula in the characteristic-function definition, $∣ Z (x) ∣ \leq \sum_{j} ∣ z_{j} ∣$ , so $∣ Z ∣^{2}$ is integrable against the probability law $μ$ . Moreover $\overline{e^{i t_{k} x}} = e^{- i t_{k} x}$ , so $∣ Z (x) ∣^{2} = \sum_{j, k} z_{j} \overline{z_{k}} e^{i (t_{j} - t_{k}) x} .$

F1F3

2.1

Integrating this finite sum gives $\sum_{j, k} z_{j} \overline{z_{k}} φ_{μ} (t_{j} - t_{k}) = \int ∣ Z (x) ∣^{2} μ (d x) \geq 0.$ The integral is real because its integrand is real and nonnegative. This verifies every quadratic form required by the definition.

step 1.1F1F2F4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Norris, Probability and Measure (standard reference, not scraped)