Central Limit Theorem theory

The Lindeberg-Lévy CLT states that if X_1, X_2, ..., X_n are independent and identically distributed random variables with finite mean μ and finite non-zero variance σ^2 (0 < σ^2 < ∞), then the standardized sample mean Z_n = (X̄_n - μ) / (σ / √n) converges in distribution to a standard normal distribution N(0, 1) as n → ∞: lim_{n→∞} P(Z_n ≤ z) = Φ(z) = (1/√2π) ∫_{-∞}^z e^(-u^2/2) du. The theorem requires only finite second moments, making no assumptions about the symmetry, continuity, or shape of the parent distribution.

Let Y_i = (X_i - μ)/σ have mean 0, variance 1, and characteristic function φ_Y(t) = E[e^(itY)]. By Taylor's theorem, φ_Y(t) = 1 - t^2/2 + o(t^2) as t → 0. The standardized sum is Z_n = (1/√n) ∑ Y_i. By independence, its characteristic function is φ_{Z_n}(t) = [φ_Y(t/√n)]^n = [1 - t^2/(2n) + o(t^2/n)]^n. Taking the limit as n → ∞: lim_{n→∞} φ_{Z_n}(t) = e^(-t^2/2). Since e^(-t^2/2) is the characteristic function of the standard normal distribution, Lévy's continuity theorem establishes that Z_n converges in distribution to N(0, 1).

The Berry-Esseen theorem (Andrew Berry 1941, Carl-Gustav Esseen 1942) quantifies the maximum uniform error between the standardized sample distribution F_n(z) and the Gaussian CDF Φ(z): sup_{z ∈ ℝ} |F_n(z) - Φ(z)| ≤ C · ρ / (σ^3 √n), where ρ = E[|X - μ|^3] is the third absolute central moment and C is a universal constant. Irina Shevtsova (2011) proved that C ≤ 0.4748. This proves that the convergence rate of the CLT is strictly O(1/√n), and that distributions with high skewness (large third moment ρ) require substantially larger sample sizes n to achieve normal approximation.

For independent random variables X_i with means μ_i, variances σ_i^2, and total variance s_n^2 = ∑_{i=1}^n σ_i^2, the Lindeberg condition requires that for every ε > 0: lim_{n→∞} (1 / s_n^2) ∑_{i=1}^n E[(X_i - μ_i)^2 · I(|X_i - μ_i| ≥ ε s_n)] = 0. This condition ensures that no single random variable dominates the total variance s_n^2. Jarl Waldemar Lindeberg (1922) proved that this condition is sufficient for the CLT; William Feller (1935) proved it is also necessary under mild conditions (Lindeberg-Feller CLT).

For a continuous uniform parent U[0, 1], the distribution is perfectly symmetric about its mean μ = 0.5, meaning its third central moment is exactly zero: μ_3 = E[(X - μ)^3] = 0. Because the skewness term in the Edgeworth expansion vanishes, the leading O(1/√n) error term in the Berry-Esseen bound drops out, leaving the remaining error to scale as O(1/n). Consequently, the sum of just n = 12 uniform variates (the classical Irwin-Hall distribution) produces a distribution that is virtually indistinguishable from a standard normal distribution.

When the parent distribution is skewed (e.g., Exponential(λ) with skewness γ_1 = 2), the third central moment μ_3 = 2/λ^3 is large and positive. The standardized distribution F_n(z) inherits this positive skewness for moderate n, resulting in an elongated right tail and a steeper left slope than the Gaussian bell curve. By the Edgeworth expansion, F_n(z) ≈ Φ(z) - (γ_1 / (6√n)) (z^2 - 1) ϕ(z). Achieving a 1% approximation accuracy for an exponential distribution requires n ≥ 30 to 50 samples, compared to n ≥ 5 for a symmetric uniform parent.

In experimental physics and engineering metrology, observed measurement errors typically arise from the superposition of numerous independent micro-scale physical disturbances (e.g., thermal fluctuations, acoustic vibrations, optical distortions, electrical noise). Under the Central Limit Theorem, the collective sum of these diverse independent error components asymptotically converges to a Gaussian distribution, providing the theoretical justification for Carl Friedrich Gauss's normal error model and least-squares analysis.