ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

The Bernoulli sample mean

Example

Let $X = {0, 1}$ with its power-set sigma-algebra. For $p \in [0, 1]$ , define the Bernoulli law $P_{p} ({1}) = p$ and $P_{p} ({0}) = 1 - p$ , and use the estimand $ψ (p) = p$ . Suppose a common probability space carries an IID sequence with one-observation law $P_{p}$ . Under its IID sample-size experiments, the estimator $T_{n} (x_{1}, \dots, x_{n}) = n^{- 1} \sum_{i = 1}^{n} x_{i}$ is the sample proportion. It satisfies the finite-sample bound $P_{p}^{\otimes n} (∣ T_{n} - p ∣ \geq ε) \leq \frac{p (1 - p)}{n ε^{2}} \leq \frac{1}{4 n ε^{2}}, ε > 0,$ so it is consistent, including at $p = 0$ and $p = 1$ .

Facts & Assumptions

Given: $p \in [0, 1]$ , the two-point observation space, the displayed Bernoulli masses, and a common probability space carrying an IID sequence with law $P_{p}$ .

[F1]

Sigma-algebras: The power set of a set is a sigma-algebra.

[F2]

Measurable spaces and measurable sets: A measurable space is a set together with a sigma-algebra.

[F3]

Measures on sigma-algebras: A measure assigns zero to the empty set and is countably additive on pairwise disjoint measurable sets.

[F4]

Probability measures and probability spaces: A probability measure is a measure of total mass one.

[F5]

Statistical experiments: An experiment is a measurable observation space and a parameter-indexed family of probability measures on it.

[F6]

Parametric statistical models: The model is the parameter map $p \mapsto P_{p}$ .

[F7]

Estimands on a statistical model: An estimand is a specified function from the parameter set to a target set.

[F8]

Estimator sequences in IID experiments: Under IID sampling, the size- $n$ law is $P_{p}^{\otimes n}$ and an estimator is a measurable map on $X^{n}$ .

[F9]

Partial sums, row sums and sample means: For real random variables, ${\overline{X}}_{n} = n^{- 1} \sum_{i = 1}^{n} X_{i}$ is the sample mean.

[F10]

Identical distribution and IID families: IID means identical laws and mutual independence.

[F11]

Independent random elements have product joint law: A finite independent family has joint law equal to the product of its marginal laws.

[F12]

The integral of a nonnegative simple function: The integral of a nonnegative simple function is the sum of its values times the measures of the corresponding disjoint level sets.

[F13]

Moments, variance, and covariance on a probability space: For square-integrable $X$ , $Var (X) = E [(X - E X)^{2}]$ .

[F14]

IID finite-variance weak law: For IID square-integrable real variables with variance $σ^{2}$ , $P (∣ {\overline{X}}_{n} - E X_{1} ∣ \geq ε) \leq σ^{2} / (n ε^{2})$ .

[F15]

Consistency of the IID sample mean under a finite first moment: An IID real sample with finite first absolute moment has a sample mean consistent in probability for its mean.

[F16]

Pointwise consistency in probability: Consistency at $p$ means the probability of an estimation error greater than each fixed $ε > 0$ tends to zero under $P_{p}^{\otimes n}$ .

[F17]

The product sigma-algebra and its finite iterates: The finite product sigma-algebra is generated by measurable rectangles, so its coordinate projections are measurable.

[F18]

Arithmetic and lattice operations preserve measurability whenever they are defined: Pointwise sums and real scalar multiples of measurable real-valued functions are measurable whenever defined.

Verification

1.1F1F2F3F4F5F6F7given

The power set of ${0, 1}$ is a sigma-algebra by [F1], so it defines a measurable space by [F2]. Because $0 \leq p \leq 1$ , the two singleton masses are nonnegative and sum to one. For any pairwise disjoint sequence of subsets, each of $0$ and $1$ occurs in at most one set, so the mass formula is countably additive and gives zero to the empty set by [F3]. Thus $P_{p}$ is a probability measure by [F4], and the family is a statistical experiment with model map $p \mapsto P_{p}$ and target $ψ (p) = p$ by [F5, F6, F7].

1.2F12F13givenalgebra

For $X \sim P_{p}$ , $X = 1_{{1}}$ and $X^{2} = X$ , so [F12] gives $E_{p} X = p$ and $E_{p} X^{2} = p$ . By [F13], ${Var}_{p} (X) = p - p^{2} = p (1 - p)$ , which is finite for all $p \in [0, 1]$ .

2.1F8F9F10F11F14F17F18step 1.2givenalgebra

By [F17], each coordinate projection on $X^{n}$ is measurable; [F18] makes their finite sum and scalar multiple $T_{n}$ measurable, as required by [F8]. The displayed map is the sample mean of [F9]. The given common sequence satisfies the IID hypothesis of [F14] by [F10], and [F11] identifies the law of its first $n$ observations with $P_{p}^{\otimes n}$ . Applying [F14] with $σ^{2} = p (1 - p)$ from step 1.2 and transferring through this joint-law identity gives $P_{p}^{\otimes n} (∣ T_{n} - p ∣ \geq ε) \leq p (1 - p) / (n ε^{2})$ . Since $p (1 - p) = 1 / 4 - (p - 1 / 2)^{2} \leq 1 / 4$ , this is at most $1 / (4 n ε^{2})$ .

3.1F8F11F15F16step 1.2step 2.1givenalgebra∎

For fixed $ε > 0$ , the bound in step 2.1 tends to zero, so [F16] gives consistency; this is also the special case of [F15]. By [F8] and [F11], at $p = 0$ and $p = 1$ the product law assigns probability one to the all-zero and all-one samples respectively, so $T_{n} = p$ almost surely at each endpoint. For the finite check $n = 2, p = 1 / 2$ , the same rectangle formula gives mass $1 / 4$ to each ordered pair, hence $T_{2}$ has masses $1 / 4, 1 / 2, 1 / 4$ at $0, 1 / 2, 1$ ; the probability of error at least $1 / 2$ is $1 / 2$ , matching the bound in step 2.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

50 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

S. R. S. Varadhan, Probability Theory, Section 3.2, PDF pp. 53–55 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., Example 2.2.4 (Bernoulli variance bound), PDF p. 65 (standard reference, not scraped)