CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

De Moivre-Laplace central limit theorem

Statement

Assume AC and fix $0 < p < 1$ . If $B_{n}$ has law $Bin (n, p)$ for every $n \geq 1$ , then $(B_{n} - n p) / \sqrt{n p (1 - p)} \Rightarrow N (0, 1) .$ No relationship between the probability spaces of the $B_{n}$ is required.

Facts & Assumptions

[F1]

Under dependent and countable choice a prescribed Bernoulli law has independent copies. Countably many independent copies of a prescribed law exist.

[F2]

AC implies dependent choice and countable choice. AC supplies countable selections and prescribed serial paths.

[F3]

Bernoulli(p) has mean p and variance p(1-p). A Bernoulli $(p)$ variable has mean $p$ and variance $p (1 - p)$ ; a binomial $(n, p)$ variable has mean $n p$ and variance $n p (1 - p)$ .

[F4]

The iid finite-positive-variance CLT applies under AC. Lindeberg-Levy iid central limit theorem.

[F5]

A binomial law is the law of a finite independent Bernoulli sum. Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials.

Proof

Given: Assume AC and fix $0 < p < 1$ . If $B_{n}$ has law $Bin (n, p)$ for every $n \geq 1$ , then $(B_{n} - n p) / \sqrt{n p (1 - p)} \Rightarrow N (0, 1) .$ No relationship between the probability spaces of the $B_{n}$ is required.

1.1

By [F2], AC supplies the countable and dependent choice required in [F1]. Apply that result to the two-point probability with masses 1-p and p to obtain iid Bernoulli variables $I_{k}$ . The law of $C_{n} = \sum_{k = 1}^{n} I_{k}$ is Bin(n,p) by [F5]. In particular it agrees with the specified law of B_n for each n; equality persists under the displayed affine standardization.

F1F2F5

2.1

By [F3], $E I_{k} = p$ and $Var (I_{k}) = p (1 - p) > 0$ . The variables are bounded, hence their second moments are finite. [F4] gives $(C_{n} - n p) / \sqrt{n p (1 - p)} \Rightarrow N (0, 1)$ . Equality of laws in step 1.1 transfers the conclusion to B_n. The excluded p=0,1 and n=0 would make the denominator zero; no claim using that denominator is made there. Neither a finite-n error estimate nor continuity correction follows from this limit theorem.

step 1.1F3F4∎

Depends on

Used by

Normal approximation to binomial probabilities Example

Dependency tree · two levels

28 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

Durrett, Probability: Theory and Examples, Section 3.1 (standard reference, not scraped)
Aldous and Chewi, Probability Theory notes, Corollary 6.1 (standard reference, not scraped)