the local central limit theorem
The ordinary CLT is a statement about distribution functions (probabilities of intervals); it says nothing directly about the DENSITY or the point probabilities. The local central limit theorem upgrades convergence in distribution to convergence of the actual density (for continuous summands) or of the point mass (for lattice summands) to the Gaussian curve, pointwise and uniformly. It is the 'pointwise' or 'pixel-level' CLT.
Two forms. Lattice case: if iid integer-valued X_i have mean mu, variance sigma^2, span 1 (gcd of the possible step differences is 1), and S_n = X_1 + ... + X_n, then as n -> infinity, sqrt(n) P(S_n = k) - (1/(sigma sqrt(2pi))) e^(-(k - n mu)^2/(2 n sigma^2)) -> 0 uniformly over integers k. Density case: if the summands have a density and the cf satisfies the integrability/decay condition integral |phi(t)|^n dt < infinity for some n (a smoothness requirement so a bounded density exists), then the density f_n of (S_n - n mu)/(sigma sqrt(n)) converges uniformly to the standard normal density (1/sqrt(2pi)) e^(-x^2/2). The proof is Fourier inversion: write the density/point-mass as an inverse transform of phi^n and show the integral concentrates near t=0 where the Gaussian approximation of the cf holds.
The local CLT matters in combinatorics and statistical physics (counting lattice paths, asymptotics of coefficients), in random walks (return probabilities, P(S_n = 0) ~ const/sqrt(n) in 1D), and anywhere you need the Gaussian approximation at the level of individual outcomes rather than aggregated probabilities. Honest caveat: it requires a non-degeneracy/smoothness condition that the global CLT does NOT. A lattice variable supported only on even integers has span 2, so P(S_n = k) = 0 for odd k and the naive local CLT fails — you must use the correct span. For continuous summands a singular or non-smooth law (cf not decaying) can satisfy the ordinary CLT yet have no limiting density at all.
For a simple symmetric random walk on the integers (steps +/-1), span is 2 since S_n has the parity of n. The correct local CLT reads P(S_2n = 2k) ~ (1/sqrt(pi n)) e^(-k^2/n) for the reachable even sites; ignoring the span-2 parity and applying the naive formula to odd sites would give nonsense.
The local CLT gives the Gaussian curve at the level of individual point probabilities — but only with the correct lattice span.
The local CLT needs a smoothness/span condition (decaying cf, or correct lattice span) beyond the ordinary CLT; a law satisfying the global CLT can have no limiting density and a wrong-span lattice formula gives zeros.