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

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Lindeberg-Levy iid central limit theorem

Statement

Assume AC. Let $(X_{k})_{k \geq 1}$ be iid real random variables with mean $m$ and variance $0 < σ^{2} < \infty$ . Then $\frac{\sum_{k = 1}^{n} X_{k} - n m}{σ \sqrt{n}} \Rightarrow N (0, 1) .$

Facts & Assumptions

[F1]

Centered variance-one variables have characteristic function 1-t^2/2+o(t^2). Second-order characteristic-function expansion.

[F2]

Near-one rows with bounded absolute sum and vanishing square sum admit exponential comparison. Products of near-one characteristic factors.

[F3]

Affine transformations and finite independent sums have the stated transform identities. Characteristic functions under affine maps and independent sums.

[F4]

Under AC the standard normal has transform exp(-t^2/2). Characteristic function of a normal law.

[F5]

Under AC convergence to the transform of a specified law gives weak convergence. Characteristic function criterion for weak convergence.

[F6]

Linearity permits centering and variance calculations. The Lebesgue integral is linear on $L^{1} (μ)$ .

Proof

Given: Assume AC. Let $(X_{k})_{k \geq 1}$ be iid real random variables with mean $m$ and variance $0 < σ^{2} < \infty$ . Then $\frac{\sum_{k = 1}^{n} X_{k} - n m}{σ \sqrt{n}} \Rightarrow N (0, 1) .$

1.1

Set $Y_{k} = (X_{k} - m) / σ$ . The affine functions are Borel; independence is preserved because an event concerning Y_k is the corresponding inverse-image event concerning X_k. Their laws agree, and linearity gives $E Y_{k} = 0$ , $E Y_{k}^{2} = 1$ . The normalized sum in the statement is $n^{- 1 / 2} \sum_{k = 1}^{n} Y_{k}$ . The positive finite variance makes every division well defined.

F6given

2.1

Fix a nonzero real t and write $w_{n} = φ_{Y_{1}} (t / \sqrt{n}) - 1$ . By [F1], $n w_{n} \to - t^{2} / 2$ . Therefore $w_{n} \to 0$ , $n ∣ w_{n} ∣$ is bounded, and $n ∣ w_{n} ∣^{2} = (n ∣ w_{n} ∣) ∣ w_{n} ∣ \to 0$ . The bounds extend to the finitely many early n since their values are finite. Apply [F2] to the row with n copies of $1 + w_{n}$ . Its product differs from $e^{n w_{n}}$ by a quantity tending to zero. Continuity of the exponential yields the limit $e^{- t^{2} / 2}$ . At t=0 each factor is exactly one.

step 1.1F1F2

3.1

By [F3], row independence identifies that product with the characteristic function of the normalized sum. By [F4] its limit is the standard-normal transform, continuous at zero and equal to one there. Thus [F5] proves the claimed convergence of laws. AC is inherited only through the normal-law and Levy-criterion suppliers; the iid sequence is given and no new copies are constructed. This theorem excludes sigma=0 because its displayed normalization divides by sigma.

step 2.1F3F4F5∎

Depends on

Used by

De Moivre-Laplace central limit theorem Corollary
CLT for sums of uniform random variables Example
Central-limit convergence is only in distribution Remark
Multivariate iid central limit theorem Theorem

Dependency tree · two levels

47 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, Theorem 3.4.1 (standard reference, not scraped)
Aldous and Chewi, Probability Theory notes, Theorem 5.2 (standard reference, not scraped)