Martingale central limit theorems replace independence by conditional centering. Their variance is accumulated predictably, so they apply to serially dependent data, likelihood scores, adaptive designs, stochastic approximation, and conditionally heteroskedastic models.
Martingale-difference arrays
Definition. A martingale-difference array is a collection \[ (X_{n,k},\mathcal F_{n,k}), \qquad1\le k\le k_n, \] such that \(\mathcal F_{n,k-1}\subseteq\mathcal F_{n,k}\), \(X_{n,k}\in L^1(\mathcal F_{n,k})\), and \[ \mathbb E[X_{n,k}\mid\mathcal F_{n,k-1}]=0. \] Its row sum and predictable variance are \[ S_n=\sum_{k=1}^{k_n}X_{n,k}, \qquad V_n=\sum_{k=1}^{k_n} \mathbb E[X_{n,k}^2\mid\mathcal F_{n,k-1}]. \]
Conditional Lindeberg theorem
Theorem (martingale CLT). Suppose \(X_{n,k}\in L^2\) form a martingale-difference array. If \[ V_n\to\sigma^2 \quad\text{in probability} \] for a constant \(\sigma^2\ge0\), and for every \(\varepsilon>0\), \[ L_n(\varepsilon) := \sum_{k=1}^{k_n} \mathbb E[ X_{n,k}^2 1_{\{|X_{n,k}|>\varepsilon\}} \mid\mathcal F_{n,k-1}] \to0 \quad\text{in probability}, \] then \[ S_n\Rightarrow N(0,\sigma^2). \]
Proof outline. Fix \(t\). Conditional Taylor expansion gives \[ \mathbb E[e^{itX_{n,k}}\mid\mathcal F_{n,k-1}] = 1-\frac{t^2}{2} \mathbb E[X_{n,k}^2\mid\mathcal F_{n,k-1}] +r_{n,k}(t), \] where, for every cutoff \(\varepsilon\), \[ \sum_k|r_{n,k}(t)| \le C t^2L_n(\varepsilon) +C|t|^3\varepsilon V_n. \] First take \(n\to\infty\), then \(\varepsilon\downarrow0\), to make the aggregate remainder vanish in probability. Unlike the independent case, the characteristic function is not literally a product of deterministic factors. One iteratively conditions backward and compares \(e^{itS_n}\) with \[ \prod_{k=1}^{k_n} \left\{1-\frac{t^2}{2} \mathbb E[X_{n,k}^2\mid\mathcal F_{n,k-1}]\right\}. \] Conditional Lindeberg implies maximal conditional variances vanish; the logarithm of this product converges in probability to \(-t^2V_n/2\to-t^2\sigma^2/2\). Uniform integrability permits taking expectations, so \[ \mathbb Ee^{itS_n}\to e^{-t^2\sigma^2/2}. \] The estimates can be localized on events where \(V_n\) and \(L_n(\varepsilon)\) are controlled; a subsequence argument converts their convergence in probability into the required expectation convergence. Levy's continuity theorem gives the result.
Corollary (conditional Lyapunov condition). If for some \(\delta>0\), \[ \sum_{k=1}^{k_n} \mathbb E[ |X_{n,k}|^{2+\delta}\mid\mathcal F_{n,k-1}] \to0 \quad\text{in probability}, \] then the conditional Lindeberg condition holds.
Proof. On \(\{|X_{n,k}|>\varepsilon\}\), \[ X_{n,k}^2 \le\varepsilon^{-\delta}|X_{n,k}|^{2+\delta}. \] Take conditional expectations and sum.
Self-normalization
Proposition. If \(V_n>0\) with probability tending to one, \(V_n\to\sigma^2>0\), and the martingale CLT assumptions hold, then \[ \frac{S_n}{\sqrt{V_n}}\Rightarrow N(0,1). \]
Proof. The CLT gives \(S_n\Rightarrow N(0,\sigma^2)\), while \(\sqrt{V_n}\to\sigma\) in probability. Apply Slutsky's theorem.
Random variance and stable convergence
Definition. Relative to a background sigma-field \(\mathcal G\), \(Y_n\) converges \(\mathcal G\)-stably to \(Y\) if \[ \mathbb E[Zf(Y_n)]\to\mathbb E[Zf(Y)] \] for every bounded \(\mathcal G\)-measurable \(Z\) and every bounded continuous \(f\). Stable convergence is stronger than weak convergence and preserves joint limits with background variables.
Theorem (mixed normal limit). Suppose the conditional Lindeberg condition holds and \(V_n\to V\) in probability for a nonnegative \(\mathcal G\)-measurable random variable \(V\). Under the corresponding conditional characteristic-function condition relative to \(\mathcal G\), \[ S_n\ \Longrightarrow_{\mathrm{st}}\ \sqrt V\,Z, \] where \(Z\sim N(0,1)\) is independent of \(\mathcal G\). In particular, \[ \mathbb Ee^{it\sqrt VZ} =\mathbb Ee^{-t^2V/2}. \]
Proof outline. Repeat the conditional characteristic-function proof after multiplying by an arbitrary bounded \(\mathcal G\)-measurable \(Z_0\). The limiting conditional characteristic function is \(e^{-t^2V/2}\). Thus \[ \mathbb E[Z_0e^{itS_n}] \to\mathbb E[Z_0e^{-t^2V/2}], \] which identifies stable convergence to a Gaussian variable conditionally on \(\mathcal G\), with conditional variance \(V\).
Predictable regression scores
Example. Let \[ Y_k=x_k^\top\beta+\varepsilon_k, \] where \(x_k\in\mathcal F_{k-1}\) is predictable, \(\mathbb E[\varepsilon_k\mid\mathcal F_{k-1}]=0\), and \(\mathbb E[\varepsilon_k^2\mid\mathcal F_{k-1}]=\sigma_k^2\). For a deterministic or predictable vector \(a_n\), the normalized score \[ S_n=\sum_{k=1}^n a_n^\top x_k\varepsilon_k \] is a martingale sum with predictable variance \[ V_n=\sum_{k=1}^n (a_n^\top x_k)^2\sigma_k^2. \] If \(V_n\to1\) and no individual score contribution is asymptotically visible, the martingale CLT gives \(S_n\Rightarrow N(0,1)\). This is the basic mechanism behind asymptotic normality with adaptive regressors and dependent but conditionally centered errors.
Example (likelihood scores). In a sequential model with conditional densities \(p_\theta(Y_k\mid\mathcal F_{k-1})\), regularity gives \[ \mathbb E_\theta[ \partial_\theta\log p_\theta(Y_k\mid\mathcal F_{k-1}) \mid\mathcal F_{k-1}]=0. \] The score is therefore a martingale sum, and its predictable bracket is the accumulated conditional Fisher information. Martingale CLTs turn convergence of that information into asymptotic normality of the score, which then feeds likelihood expansions in local asymptotic normality and contiguity.