Definition D1
X₁, …, Xₙ are mutually independent and identically distributed: E(Xᵢ) = μ, Var(Xᵢ) = σ² > 0. Define Sₙ = X₁ + ⋯ + Xₙ and Zₙ = (Sₙ − nμ)/(σ√n).
Proof.
(1)Define the variance by Var(X) = E[(X − E X)²] and the covariance by Cov(X, Y) = E[(X − E X)(Y − E Y)].convention
(2)Zₙ is Sₙ with its expectation subtracted, divided by σ√n. That these two constants are chosen correctly is the content of T1.convention
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E(Sₙ) = E(X₁) + ⋯ + E(Xₙ) = nμD1, A1, N
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Lemma L2
Var(Sₙ) = nσ²
Proof.
(1)Sₙ − nμ = Σ(Xᵢ − μ)D1, L1, N
(2)(Sₙ − nμ)² = ΣᵢΣⱼ (Xᵢ − μ)(Xⱼ − μ)(1), A3
(3)Var(Sₙ) = E[(Sₙ − E Sₙ)²] = ΣᵢΣⱼ E[(Xᵢ−μ)(Xⱼ−μ)]D1, L1, (2), A1
(4)For i ≠ j, Cov(Xᵢ, Xⱼ) = 0. All n(n−1) such terms vanish.D1, A2
(5)For i = j, E[(Xᵢ−μ)²] = σ². All n such terms remain.D1
(6)Var(Sₙ) = nσ²(3), (4), (5), N
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Theorem T1
For every n, E(Zₙ) = 0, Var(Zₙ) = 1
Proof.
(1)E(Zₙ) = (E(Sₙ) − nμ)/(σ√n) = 0D1, L1, A1, N
(2)Zₙ − E(Zₙ) = (Sₙ − nμ)/(σ√n), so Var(Zₙ) = E[(Sₙ−nμ)²]/(σ²n) = Var(Sₙ)/(σ²n)D1, (1), A1, N
(3) = nσ²/(σ²n) = 1(2), L2, N
(4)This equality holds at each finite n, not in the limit. It is fully proved from the axioms of this list.(1), (3)
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Statement T2
As n grows, the distribution of Zₙ approaches the standard normal N(0,1): for every real z, P(Zₙ ≤ z) → Φ(z).
(1)Φ is the distribution function of N(0,1). The proposition is convergence of the distribution function at each point z.convention
(2)The proof of this convergence requires characteristic functions or moment tools. The axioms of this list do not reach it, so here it is left as a statement without proof.convention
(3)The premises of the statement include independence. Without independence this convergence itself fails.D1, A2
Proof omitted.
Corollary C
That the mean and variance are right, and that the distribution goes normal, are claims on different levels.
Proof.
(1)T1 is an equality holding at each finite n, fully proved from the axioms of this list.T1
(2)T2 is convergence of distributions as n → ∞, a statement requiring tools beyond this list.T2
(3)T1 fixes only the first two moments of Zₙ; it does not determine the distribution. There are countless distributions with mean 0 and variance 1.T1, N
(4)That among those countless possibilities everything gathers to the one normal distribution is the content of T2. Here lies the boundary between what is proved and what is stated.(3), T2
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