A triangular array lets the number and distribution of summands vary with the asymptotic index. This is the correct language for non-identically distributed central limit theorems, rare-event limits, local alternatives, bootstrap weights, and asymptotically negligible remainders. Truncation supplies a common proof architecture: isolate large jumps, center the small part, and identify the limiting characteristic exponent.
Rows, infinitesimality, and normalization
Definition. A triangular array is a collection \[ \{X_{n,k}:1\le k\le k_n,\ n\ge1\}. \] It is row-wise independent if \(X_{n,1},\ldots,X_{n,k_n}\) are independent for each fixed \(n\). It is uniformly asymptotically negligible, or infinitesimal, if for every \(\varepsilon>0\), \[ \max_{1\le k\le k_n}P(|X_{n,k}|>\varepsilon)\longrightarrow0. \]
Infinitesimality says that no individual summand remains visible at the scale of the row sum. It does not determine the limit: aggregated small Gaussian fluctuations, rare jumps, or both may survive.
Proposition (Feller condition implies infinitesimality). Suppose \(X_{n,k}\) are centered and square-integrable, and \[ \max_{k\le k_n}\operatorname{Var}(X_{n,k})\longrightarrow0. \] Then the array is infinitesimal.
Proof. Chebyshev's inequality gives \[ \max_{k\le k_n}P(|X_{n,k}|>\varepsilon) \le\varepsilon^{-2}\max_{k\le k_n}\operatorname{Var}(X_{n,k})\to0. \]
Lindeberg-Feller convergence
Definition. For a centered, row-wise independent array, put \[ s_n^2=\sum_{k=1}^{k_n}\mathbb EX_{n,k}^2. \] The Lindeberg condition at the row scale \(s_n\) is \[ L_n(\varepsilon) :=\frac1{s_n^2}\sum_{k=1}^{k_n} \mathbb E\!\left[X_{n,k}^2 1_{\{|X_{n,k}|>\varepsilon s_n\}}\right]\longrightarrow0 \] for every \(\varepsilon>0\).
Theorem (Lindeberg-Feller). Let \(X_{n,k}\) be centered and row-wise independent. If \(s_n^2\to\sigma^2\in(0,\infty)\) and the Lindeberg condition holds, then \[ \sum_{k=1}^{k_n}X_{n,k}\Rightarrow N(0,\sigma^2). \] Conversely, if the row sums converge to \(N(0,\sigma^2)\) and the Feller condition \[ \max_{k\le k_n}\frac{\operatorname{Var}(X_{n,k})}{s_n^2}\to0 \] holds, then the Lindeberg condition holds.
Proof of sufficiency. Rescale to \(s_n^2\to1\). Let \(\varphi_{n,k}(t)=\mathbb Ee^{itX_{n,k}}\). Centering gives \[ \varphi_{n,k}(t) =1-\frac{t^2}{2}\mathbb EX_{n,k}^2+r_{n,k}(t). \] For every \(\delta>0\), Taylor's formula and \(|e^{ix}-1-ix+x^2/2|\le \min\{C|x|^2,\,|x|^3/6\}\) imply \[ \sum_k|r_{n,k}(t)| \le C t^2\sum_k\mathbb E[X_{n,k}^2 1_{\{|X_{n,k}|>\delta\}}] +C|t|^3\delta\sum_k\mathbb EX_{n,k}^2. \] The first term tends to zero by Lindeberg; after taking \(n\to\infty\), the second can be made arbitrarily small by \(\delta\downarrow0\). Thus \[ \sum_k r_{n,k}(t)\to0. \tag{1} \] Lindeberg also implies the Feller condition because \[ \mathbb EX_{n,k}^2 \le \delta^2+ \mathbb E[X_{n,k}^2 1_{\{|X_{n,k}|>\delta\}}], \] uniformly in \(k\). Hence \(\max_k|\varphi_{n,k}(t)-1|\to0\). Using \(\log(1+z)=z+O(|z|^2)\), (1), and \(\sum_k\mathbb EX_{n,k}^2\to1\), \[ \log\prod_k\varphi_{n,k}(t) =-\frac{t^2}{2}\sum_k\mathbb EX_{n,k}^2+o(1) \longrightarrow-\frac{t^2}{2}. \] Row-wise independence makes the product the characteristic function of the row sum. Levy's continuity theorem proves convergence to \(N(0,1)\).
Proof outline of necessity under Feller's condition. Compare the row sum with a Gaussian row \(Z_{n,k}\sim N(0,\operatorname{Var}(X_{n,k}))\), independent across \(k\). A one-at-a-time replacement in the characteristic function shows that normal convergence of the total sum, together with \(\max_k\operatorname{Var}(X_{n,k})\to0\), forces the aggregate second moment carried by \(|X_{n,k}|>\varepsilon\) to vanish. Equivalently, the Levy measures of the infinitesimal array have no mass away from zero. This is precisely \(L_n(\varepsilon)\to0\). The full replacement estimate is the converse part of the Lindeberg theorem; it uses the same second-order Taylor remainder as the sufficiency proof, now read in reverse.
Lyapunov's moment condition
Corollary (Lyapunov). If, for some \(\delta>0\), \[ \frac1{s_n^{2+\delta}}\sum_{k=1}^{k_n} \mathbb E|X_{n,k}|^{2+\delta}\longrightarrow0, \] then the Lindeberg condition holds and the normalized row sums converge to \(N(0,1)\).
Proof. On \(\{|X_{n,k}|>\varepsilon s_n\}\), \[ X_{n,k}^2\le \frac{|X_{n,k}|^{2+\delta}}{(\varepsilon s_n)^\delta}. \] Summing and dividing by \(s_n^2\) gives \[ L_n(\varepsilon) \le\varepsilon^{-\delta}s_n^{-(2+\delta)} \sum_k\mathbb E|X_{n,k}|^{2+\delta}\to0. \]
Truncation and transfer of limits
Lemma (converging together). If \(T_n-U_n\to0\) in probability and \(U_n\Rightarrow U\), then \(T_n\Rightarrow U\).
Proof. The pair \((U_n,T_n-U_n)\Rightarrow(U,0)\) by Slutsky's theorem, and addition is continuous.
Proposition (tail deletion). Let \[ Y_{n,k}=X_{n,k}1_{\{|X_{n,k}|\le a_n\}}. \] Either of the conditions \[ \sum_{k=1}^{k_n}P(|X_{n,k}|>a_n)\to0 \] or \[ \sum_{k=1}^{k_n} \mathbb E[|X_{n,k}|1_{\{|X_{n,k}|>a_n\}}]\to0 \] implies \(\sum_k(X_{n,k}-Y_{n,k})\to0\) in probability.
Proof. Under the first condition, \[ P\!\left(\exists k:X_{n,k}\ne Y_{n,k}\right) \le\sum_kP(|X_{n,k}|>a_n)\to0. \] Under the second, Markov's inequality gives, for every \(\varepsilon>0\), \[ P\!\left(\sum_k|X_{n,k}-Y_{n,k}|>\varepsilon\right) \le\varepsilon^{-1}\sum_k \mathbb E[|X_{n,k}|1_{\{|X_{n,k}|>a_n\}}]\to0. \] The converging-together lemma then transfers any weak limit proved for the truncated sum.
Infinitesimal arrays need not be Gaussian
Example (rare jumps). Let \(X_{n,1},\ldots,X_{n,n}\) be independent Bernoulli\((\lambda/n)\). The array is infinitesimal because \[ \max_{k\le n}P(|X_{n,k}|>\tfrac12)=\frac{\lambda}{n}\to0, \] but \[ \sum_{k=1}^nX_{n,k}\Rightarrow\operatorname{Poisson}(\lambda), \] not a Gaussian law. Here the Lindeberg condition fails at the unscaled jump size \(1\): rare jumps retain a nonvanishing aggregate intensity.
The general limits of infinitesimal row sums are exactly the infinitely divisible laws. The Gaussian theorem corresponds to the case in which all limiting jump mass disappears; Levy-Khintchine theory records what remains when it does not.