LindebergFeller

The Lindeberg–Feller theorem is a central limit theorem for sums of independent random variables that are not necessarily identically distributed. It is commonly stated for triangular arrays, where for each n there are independent random variables X_{n,1}, ..., X_{n,N_n} with means zero and finite variances, and the sum S_n = sum_{k=1}^{N_n} X_{n,k} is considered.

Let s_n^2 = Var(S_n) = sum_{k=1}^{N_n} Var(X_{n,k}). If s_n^2 tends to infinity (or to a finite positive limit σ^2 in a variant) and the Lindeberg condition holds, then the normalized sum S_n / s_n converges in distribution to a normal random variable. The Lindeberg condition requires that for every ε > 0, (1/s_n^2) sum_{k=1}^{N_n} E[ X_{n,k}^2 I{|X_{n,k}| > ε s_n} ] → 0 as n → ∞. This condition ensures that no single term dominates the sum in the limit.

The theorem generalizes the classical central limit theorem to independent, non-identically distributed variables. A common alternative sufficient condition is the Lyapunov condition, which uses higher moments. The Lindeberg–Feller theorem, named after J. Lindeberg and W. Feller, has wide applications in probability theory and statistics, including econometrics and reliability theory.