the Levy-Khintchine formula

The Lévy-Khintchine formula is the complete classification of infinitely divisible laws via their characteristic functions: it answers exactly which functions psi can serve as the logarithm of an infinitely divisible characteristic function, and it does so by splitting psi into three transparent pieces — a drift, a Gaussian curvature, and a jump integral. It is the structural backbone of the whole theory: it converts the analytically awkward convolution condition of infinite divisibility into a clean, parametrized, geometric description.

The theorem states: phi(theta) = E[e^(i theta X)] is the characteristic function of an infinitely divisible law on R if and only if phi(theta) = exp(psi(theta)) with psi(theta) = i b theta - (1/2) sigma^2 theta^2 + integral over R\{0} of (e^(i theta x) - 1 - i theta x 1_{|x|<1}) nu(dx). Here the triplet (b, sigma^2, nu) — drift b in R, Gaussian variance sigma^2 >= 0, and Lévy measure nu on R\{0} with integral of min(1, x^2) nu(dx) < infinity — is unique. The compensator term -i theta x 1_{|x|<1} is exactly what is needed to make the integral converge near zero when small jumps accumulate; large jumps (|x| >= 1) are summable on their own. The cutoff at 1 is a convention: changing it only shifts b.

The formula explains the parameters of every named process: Brownian motion with drift is (b, sigma^2, 0); the compound Poisson process is (0, 0, lambda F) with F a probability law; the gamma process and stable processes have explicit nu. It also gives the generator of the associated Lévy process as a pseudo-differential operator with symbol psi. A standard pitfall: the integrand and the cutoff must travel together — quoting psi with the e^(i theta x) - 1 integrand but no compensator is only legitimate when integral of min(1,|x|) nu(dx) < infinity (finite-variation jumps), otherwise the integral diverges.