CLT

Central Limit Theorem (CLT) is a fundamental result in probability theory. It states that the sum or average of a large number of independent, identically distributed random variables with finite mean and variance tends to be normally distributed, regardless of the original distribution of the variables.

Formally, if X1, X2, ..., Xn are iid with mean μ and variance σ^2, and S_n = ∑ X_i, then (S_n - n μ) / (√n σ) converges in distribution to standard normal N(0,1) as n → ∞. Equivalently, the sample mean X̄_n = S_n/n satisfies (X̄_n - μ) / (σ/√n) → N(0,1).

The theorem requires independence and identical distribution with finite mean and variance; many variants relax these conditions, such as Lindeberg-Feller and Lyapunov central limit theorems, which apply to arrays of independent but not identically distributed variables, under moment and Lindeberg conditions.

Consequences: The CLT justifies the use of normal approximations for sums and averages of random variables in statistics. It underpins confidence intervals and hypothesis testing in large samples, and informs sampling distributions and standard error estimates. Rates of convergence can be characterized by the Berry-Esseen theorem, which provides a bound on the difference between the distribution of the standardized sum and the standard normal, typically of order O(1/√n) under certain moment conditions.

History: Early forms were noted by de Moivre and Laplace in the context of binomial probabilities; the modern CLT was developed by Lyapunov and later Lindeberg in the early 20th century.