Central limit theorems describe the second-order behavior left after a law of large numbers removes deterministic drift. Finite variance selects the scale \(\sqrt n\); asymptotic negligibility makes the quadratic part of the characteristic exponent universal, and the resulting limit is Gaussian.
Centering and fluctuation scale
Definition. For i.i.d. variables with \(\mathbb EX_1=\mu\) and \(0<\operatorname{Var}(X_1)=\sigma^2<\infty\), define \[ Z_n=\frac{S_n-n\mu}{\sigma\sqrt n} =\frac{\sqrt n(\overline X_n-\mu)}{\sigma}. \] The law of large numbers says \(\overline X_n-\mu=o_P(1)\); a central limit theorem identifies the nondegenerate limit of the magnified error \(\sqrt n(\overline X_n-\mu)\).
Lemma (second-order characteristic expansion). If \(\mathbb EX=0\) and \(\mathbb EX^2=\sigma^2<\infty\), then, as \(t\to0\), \[ \varphi_X(t)=\mathbb Ee^{itX} =1-\frac{\sigma^2t^2}{2}+o(t^2). \]
Proof. Write \[ \frac{e^{itX}-1-itX}{t^2} =X^2\,\frac{e^{iu}-1-iu}{u^2}\bigg|_{u=tX}. \] The scalar factor tends to \(-1/2\) as \(u\to0\) and is bounded in absolute value by a universal constant, since \(|e^{iu}-1-iu|\le u^2/2\). Dominated convergence, with dominating variable \(X^2/2\), gives \[ \frac{\varphi_X(t)-1-it\mathbb EX}{t^2}\to-\frac12\mathbb EX^2. \] Centering removes the linear term.
The classical central limit theorem
Theorem (Lindeberg-Levy). If \(X_1,X_2,\ldots\) are i.i.d. with mean \(\mu\) and variance \(0<\sigma^2<\infty\), then \[ \frac{S_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1). \]
Proof. Standardize so that \(\mu=0\) and \(\sigma^2=1\), and let \(\varphi\) be the characteristic function of \(X_1\). The lemma gives \[ \varphi(t/\sqrt n) =1-\frac{t^2}{2n}+o(n^{-1}). \] Independence yields \[ \mathbb E e^{itS_n/\sqrt n} =\varphi(t/\sqrt n)^n. \] Since \(n\log(1+u_n)=nu_n+o(1)\) whenever \(u_n=O(n^{-1})\), \[ n\log\varphi(t/\sqrt n)\longrightarrow-\frac{t^2}{2}. \] Thus the characteristic functions converge pointwise to \(e^{-t^2/2}\), which is continuous at zero and is the characteristic function of \(N(0,1)\). Levy's continuity theorem completes the proof.
Example (de Moivre-Laplace). If \(B_n\sim\operatorname{Binomial}(n,p)\), then \[ \frac{B_n-np}{\sqrt{np(1-p)}}\Rightarrow N(0,1). \] Consequently, for fixed \(a<b\), \[ P\!\left( a\le\frac{B_n-np}{\sqrt{np(1-p)}}\le b \right)\to\Phi(b)-\Phi(a). \] A continuity correction improves finite-\(n\) approximation because the binomial law is supported on a lattice.
Non-identical summands
Theorem (Lindeberg-Feller, normalized form). Let \(X_{n,k}\) be centered and row-wise independent, with \[ \sum_{k=1}^{k_n}\operatorname{Var}(X_{n,k})\to1. \] If, for every \(\varepsilon>0\), \[ \sum_{k=1}^{k_n} \mathbb E[X_{n,k}^2 1_{\{|X_{n,k}|>\varepsilon\}}]\to0, \] then \(\sum_kX_{n,k}\Rightarrow N(0,1)\).
The detailed characteristic-function proof and the converse under Feller's condition are given on the preceding triangular-array page. The condition says that asymptotically none of the total variance is carried by jumps visible at the scale of the whole sum.
Corollary (Lyapunov CLT). Let \(X_1,X_2,\ldots\) be independent and centered, put \(s_n^2=\sum_{i=1}^n\mathbb EX_i^2\), and assume \(s_n\to\infty\). If for some \(\delta>0\), \[ \frac1{s_n^{2+\delta}}\sum_{i=1}^n \mathbb E|X_i|^{2+\delta}\to0, \] then \(S_n/s_n\Rightarrow N(0,1)\).
Multivariate Gaussian limits
Theorem (Cramer-Wold device). Random vectors \(Y_n,Y\in\mathbb R^d\) satisfy \(Y_n\Rightarrow Y\) if and only if \[ t^\top Y_n\Rightarrow t^\top Y \qquad\text{for every }t\in\mathbb R^d. \]
Proof. The forward implication follows from the continuous mapping theorem. Conversely, \[ \varphi_{Y_n}(t)=\mathbb Ee^{it^\top Y_n} \longrightarrow\mathbb Ee^{it^\top Y} =\varphi_Y(t) \] for each \(t\). Multivariate Levy continuity gives \(Y_n\Rightarrow Y\).
Theorem (multivariate i.i.d. CLT). If \(X_i\in\mathbb R^d\) are i.i.d., \(\mathbb E\|X_1\|^2<\infty\), \(\mathbb EX_1=\mu\), and \(\operatorname{Cov}(X_1)=\Sigma\), then \[ \sqrt n(\overline X_n-\mu)\Rightarrow N_d(0,\Sigma). \] The covariance matrix may be singular.
Proof. For each \(t\in\mathbb R^d\), the scalar variables \(t^\top(X_i-\mu)\) are i.i.d., centered, and have variance \(t^\top\Sigma t\). The scalar CLT gives \[ t^\top\sqrt n(\overline X_n-\mu) \Rightarrow N(0,t^\top\Sigma t). \] These are exactly the one-dimensional projections of \(N_d(0,\Sigma)\); apply Cramer-Wold.
Studentization and smooth transformations
Corollary (Studentized mean). Under the scalar i.i.d. assumptions, let \[ \widehat\sigma_n^2=\frac1n\sum_{i=1}^n(X_i-\overline X_n)^2. \] Then \[ \frac{\sqrt n(\overline X_n-\mu)}{\widehat\sigma_n} \Rightarrow N(0,1). \]
Proof. The strong law applied to \(X_i\) and \(X_i^2\) gives \[ \widehat\sigma_n^2 =\frac1n\sum_iX_i^2-\overline X_n^2\to \mathbb EX_1^2-\mu^2=\sigma^2 \] almost surely. Hence \(\sigma/\widehat\sigma_n\to1\) in probability, and Slutsky's theorem combines this with the CLT.
Proposition (delta method). If \(\sqrt n(T_n-\theta)\Rightarrow Z\) in \(\mathbb R^d\) and \(g:\mathbb R^d\to\mathbb R^m\) is differentiable at \(\theta\), then \[ \sqrt n\{g(T_n)-g(\theta)\} \Rightarrow Dg(\theta)Z. \]
Proof. Differentiability gives \[ g(\theta+h)-g(\theta)=Dg(\theta)h+r(h), \qquad \frac{\|r(h)\|}{\|h\|}\to0. \] Tightness of \(\sqrt n(T_n-\theta)\) implies \(T_n-\theta=O_P(n^{-1/2})\), so \(\sqrt n\,r(T_n-\theta)=o_P(1)\). The conclusion follows from Slutsky's theorem.
The delta method is developed further in the asymptotic tools for statistical inference, including higher-order cases where \(Dg(\theta)=0\).
Boundaries of Gaussian approximation
Remark. A CLT is a statement about fixed, standardized distributional regions. It does not automatically approximate probabilities far into a moving tail, nor does it provide a convergence rate. Berry-Esseen bounds address rates under a third moment, while moderate- and large-deviation theory governs tail regions.
Example (infinite variance). If \(X_i\) have regularly varying tails with index \(\alpha\in(0,2)\), the normalization \(\sqrt n\) is generally wrong. Under suitable balance conditions, \[ \frac{S_n-b_n}{a_n}\Rightarrow Z_\alpha, \qquad a_n\asymp n^{1/\alpha}L(n), \] where \(Z_\alpha\) is an \(\alpha\)-stable law. Gaussian universality is therefore a finite-variance phenomenon, not a universal law for every sum.