Exponential decay of tail probabilities. Visualize the cumulant generating function, the Legendre-Fenchel transform giving the rate function, and verify P(S_n/n ≥ x) ~ exp(-n·I(x)) with Monte Carlo.
Cramér (1938): P(S_n/n ≥ x) ~ e^{-n·I(x)} where I(x) = sup_λ{λx - Λ(λ)} (Legendre-Fenchel transform of CGF Λ(λ)=log M_X(λ)). The saddle-point approximation gives the dominating exponential. Varadhan's lemma: for continuous bounded f, lim_{n→∞} (1/n) log E[e^{n·f(S_n/n)}] = sup_x{f(x)-I(x)}. Gärtner-Ellis: works for non-iid with limits of CGFs. Entropy connection: I = relative entropy D_KL for exponential families.