the general central limit problem
The classical CLT (Gaussian limit), the generalized CLT (stable limits), and the Poisson limit are all special cases of one grand question, posed and largely solved by Levy, Khintchine, Gnedenko and Kolmogorov: for a triangular array of independent, individually negligible summands, what are ALL possible limit laws of the row sums, and exactly when does convergence to a given limit occur? This is the general central limit problem, and its answer organises classical limit theory.
Take a triangular array {X_{n,k}} that is uniformly asymptotically negligible (max_k P(|X_{n,k}| > epsilon) -> 0). The first structural theorem: the set of possible (non-degenerate) limit laws of the centered row sums is EXACTLY the class of infinitely divisible distributions — those whose cf can be written via the Levy-Khintchine formula log phi(t) = i b t - sigma^2 t^2/2 + integral ( e^(itx) - 1 - itx 1{|x|<=1} ) nu(dx), with a Gaussian variance sigma^2 and a Levy measure nu controlling jumps. The second: convergence of the row sums to a particular infinitely divisible law happens if and only if the array's accompanying objects converge — the sum of truncated variances tends to the Gaussian part, and the array's empirical jump measure converges (vaguely, away from 0) to the Levy measure nu. The cf method is the proof engine: one shows the log-cf of the row sum converges to the Levy-Khintchine exponent and applies Levy's continuity theorem.
The general central limit problem matters as the unifying summit of the subject: the Gaussian (nu=0), the stable laws (a scale-invariant nu), the Poisson (nu a point mass), and compound Poisson laws all drop out as instances by choosing the Gaussian part and Levy measure. It is also the static, one-time-slice shadow of Levy processes (where the same Levy-Khintchine triple generates a whole process). Honest caveats: the clean infinitely-divisible classification requires the negligibility hypothesis — drop it and a single dominant term can produce a non-infinitely-divisible limit. And this is the distributional limit problem for SUMS; it is distinct from (though deeply parallel to) the Levy-Khintchine description of processes, which lives in the Levy-process field, not here.
Three classic outcomes of the same machinery: rows of tiny Bernoulli(lambda/n) terms give a Poisson limit (Levy measure = point mass at 1); rows of finite-variance terms with negligible large values give a Gaussian limit (Levy measure = 0); rows with regularly-varying tails give a stable limit (scale-invariant Levy measure). The choice of Gaussian part and Levy measure selects which.
All limits of negligible-row triangular arrays are infinitely divisible, indexed by a Gaussian part and a Levy measure.
The infinitely-divisible classification holds only under uniform asymptotic negligibility; this is the limit problem for SUMS, distinct from the Levy-Khintchine description of whole processes.