a triangular array of random variables
Most limit theorems beyond the iid CLT are most cleanly stated not for a single sequence but for an ARRAY whose rows change with n. The triangular array is the right bookkeeping device: for each n you have a fresh row of summands, possibly of different length and different distributions, and you sum the n-th row. It is the natural home of the Lindeberg-Feller theorem and the general central limit problem.
A triangular array is a doubly-indexed collection {X_{n,k} : 1 <= k <= r_n}, n = 1, 2, ..., where the variables WITHIN each row n are independent (rows for different n need not be related at all). One studies the row sums S_n = sum_{k=1}^{r_n} X_{n,k}. The crucial feature that makes the framework powerful is that the distribution of X_{n,k} can depend on n: as n grows, each individual summand can become smaller (its variance shrinking) while the number of summands grows, so that the row sum has a non-trivial limit. This captures situations a fixed sequence cannot, for instance normalized sums (X_{n,k} = X_k/(sigma sqrt(n))) where the array entries themselves are rescaled. A standard regularity demand is uniform asymptotic negligibility: max_k P(|X_{n,k}| > epsilon) -> 0 for every epsilon, meaning no single term in a row is macroscopic.
Triangular arrays matter because the deepest classical limit theory lives here: under uniform asymptotic negligibility, the only possible limits of row sums are the infinitely divisible laws (which include the Gaussian and the stable laws), and the precise convergence conditions are the array versions of Lindeberg/Lyapunov, or the convergence of the array's accompanying Levy measures and Gaussian part. Honest caveats: the negligibility hypothesis is what restricts the limits to the infinitely-divisible class — without it almost any limit can occur. And the convergence depends on the WHOLE array, not the limit of any fixed row; you must control the joint behaviour as both r_n grows and the entry distributions change.
The Poisson limit is a triangular array result: let X_{n,k} be independent Bernoulli(lambda/n) for k=1,...,n. Each term is negligible (P(X_{n,k}=1)=lambda/n -> 0), the row sum counts successes, and S_n = sum_k X_{n,k} -> Poisson(lambda) in distribution — a non-Gaussian infinitely-divisible limit obtained precisely because the array entries shrink as n grows.
Triangular arrays let each summand shrink with n; under negligibility the limits are exactly the infinitely divisible laws.
Uniform asymptotic negligibility (no single row term is macroscopic) is what confines the possible limits to the infinitely divisible class; drop it and almost any limit law can appear.