ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

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A predictable linear forecast and its empirical residual process

Example

Assume AC, fix $δ \in (0, 1)$ , and choose a Borel one-sided sub-exponential uniform boundary $u_{δ / 2, 1}$ with crossing probability $δ / 2$ at scale $1$ . Let $X_{1}, X_{2}, \dots$ be independent Bernoulli random variables with success probability $1 / 4$ , set $X_{0} = 0$ , and let $F_{0} = {\emptyset, Ω}$ and $F_{t} = σ (X_{1}, \dots, X_{t})$ . Use the predictable linear forecast ${\hat{X}}_{t} = 1 / 4 + (1 / 2) X_{t - 1}$ . Then ${\hat{X}}_{t} \in [0, 1]$ and $E [X_{t} ∣ F_{t - 1}] = 1 / 4$ almost surely. On the path $(X_{1}, X_{2}, X_{3}, X_{4}) = (1, 0, 0, 1)$ , which has probability $9 / 256$ , the algorithm has ${\overset{ˉ}{X}}_{4} = 1 / 2$ , $V_{4} = 7 / 4$ , and returns $C_{4} = [\frac{1}{2} - \frac{u_{δ / 2, 1} (7 / 4)}{4}, \frac{1}{2} + \frac{u_{δ / 2, 1} (7 / 4)}{4}] .$ The forecast is predictable and differs from the conditional mean when $X_{t - 1} = 1$ ; the full interval sequence still covers its target sequence with probability at least $1 - δ$ .

Example references

The one-sided boundary input and its crossing guarantee

The streaming interval update

The all-times coverage theorem

Facts & Assumptions

Given: Assume AC. The Bernoulli stream, filtration, forecast, error level, boundary, algorithm, and target sequence in the Example.

[F1]

Conditional expectation given a sigma algebra: a conditional-expectation version is measurable and has the defining event integrals.

[F2]

Conditional expectation is unique almost surely: conditional-expectation versions with the same input and conditioning sigma-algebra agree almost surely.

[F3]

Empirical-Bernstein confidence-sequence interval update: at time $t$ , the output is centered at ${\overset{ˉ}{X}}_{t}$ with radius $u_{δ / 2, 1} (V_{t}) / t$ .

[F4]

Empirical-Bernstein confidence sequence under adaptive sampling: for these bounded observations and predictable forecasts, the intervals cover the running average conditional-mean target at all times with probability at least $1 - δ$ .

[A1]

The Axiom of Choice: AC is explicitly assumed here and by the coverage theorem; the definition of a conditional-expectation version imposes no AC assumption.

Verification

1.1givenF1F2

For each $t$ , the atoms of $F_{t - 1}$ are the events $A_{x} = {(X_{1}, \dots, X_{t - 1}) = x}$ , with $A = Ω$ for $t = 1$ . Each atom has positive probability, and independence gives $P (X_{t} = 1, A_{x}) = \frac{1}{4} P (A_{x})$ and $P (X_{t} = 0, A_{x}) = \frac{3}{4} P (A_{x})$ . Thus the constant $1 / 4$ has the defining event integrals on every atom and every union of atoms, so it is a version of $E [X_{t} ∣ F_{t - 1}]$ by [F1]; [F2] identifies it with the chosen conditional-expectation class.

2.1givenstep 1.1algebra

The forecast ${\hat{X}}_{t}$ is a function of $X_{t - 1}$ , hence $F_{t - 1}$ -measurable, and its values are $1 / 4$ or $3 / 4$ ; when $X_{t - 1} = 1$ it equals $3 / 4$ , not the conditional mean $1 / 4$ . On the specified path, the four squared errors are $9 / 16, 9 / 16, 1 / 16, 9 / 16$ , so $V_{4} = 28 / 16 = 7 / 4$ and ${\overset{ˉ}{X}}_{4} = 2 / 4 = 1 / 2$ . Independence gives the path probability $(1 / 4) (3 / 4) (3 / 4) (1 / 4) = 9 / 256$ .

3.1givenF3F4A1step 1.1algebra∎

By [F3], the algorithm returns the displayed interval at time $4$ . Its hypotheses hold with $a = 0$ , $b = 1$ ; therefore [F4] gives $P (μ_{t} \in C_{t} for every t \geq 1) \geq 1 - δ$ . In particular, using [A1], the interval sequence is a confidence sequence for the adapted target process defined in the Example.

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