ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

A finite-horizon likelihood-ratio test supermartingale

Example

Fix an integer $K \geq 1$ . Let $Ω = {0, 1}^{K}$ and let $P_{0}$ be uniform on $Ω$ , so the coordinate variables $X_{1}, \dots, X_{K}$ are independent Bernoulli $(1 / 2)$ . Let $Q$ be the product Bernoulli $(3 / 4)$ law and let $F_{t} = σ (X_{1}, \dots, X_{\min (t, K)})$ . The likelihood ratio of $Q$ to $P_{0}$ on $F_{t}$ is $E_{t} = \prod_{i = 1}^{\min (t, K)} ℓ (X_{i}), ℓ (1) = \frac{3}{2}, ℓ (0) = \frac{1}{2},$ with the empty product $E_{0} = 1$ . It is a nonnegative test martingale, hence a nonnegative test supermartingale, for the simple null ${P_{0}}$ .

Facts & Assumptions

Given: $K \geq 1$ , the finite sample space, laws, filtration, and process in the Example. Assume AC for the test-supermartingale convention.

[F1]

Nonnegative test supermartingales: the process must be real-valued, adapted, integrable, nonnegative, supermartingale under the null, and have initial expectation at most one.

[A1]

The Axiom of Choice: AC supplies the conditional-expectation convention used in F1.

Verification

1.1givenconstruct

For a prefix $x_{1}, \dots, x_{t}$ with $t \leq K$ , the ratio of its $Q$ -probability to its $P_{0}$ -probability is $\prod_{i = 1}^{t} (Q (X_{i} = x_{i}) / P_{0} (X_{i} = x_{i})) = \prod_{i = 1}^{t} ℓ (x_{i})$ , because every $P_{0}$ prefix has positive probability. Thus the displayed process is the finite-prefix likelihood ratio; after time $K$ it is held constant.

2.1givenstep 1.1algebra

Each $E_{t}$ is a finite nonnegative function of the first $\min (t, K)$ coordinates, hence is adapted and integrable. Conditional on any prefix before time $K$ , the next bit is fair and $E_{P_{0}} [ℓ (X_{t + 1}) ∣ F_{t}] = (3 / 2 + 1 / 2) / 2 = 1$ , so $E_{P_{0}} [E_{t + 1} ∣ F_{t}] = E_{t}$ . For $t \geq K$ the process is constant.

3.1F1A1step 1.1step 2.1∎

Since $E_{0} = 1$ , steps 1.1 and 2.1 verify every condition in F1 under the null law $P_{0}$ ; hence $E$ is a test martingale and a test supermartingale.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.