the Levy measure
/ lev-EE /
The Lévy measure is the bookkeeping device for jumps: it records, per unit time, the expected number of jumps falling in each region of sizes. It is the third and richest component of the Lévy-Khintchine triplet, and it alone distinguishes the pure-jump processes from one another. Where the Gaussian part sets the diffusive wiggle, the Lévy measure sets the entire jump structure.
Formally nu is a measure on R\{0} satisfying the integrability condition integral of min(1, x^2) nu(dx) < infinity; this is exactly what makes the Lévy-Khintchine integral converge. Its meaning is operational: for a Borel set A bounded away from 0, the number of jumps of X with size in A up to time t is Poisson distributed with mean t nu(A), and disjoint size-regions give independent jump counts. Thus nu(A) is the rate of jumps of size in A. Near zero nu may be infinite (infinitely many tiny jumps per unit time), but the x^2 weight keeps their squared contribution summable; away from zero nu is finite if the process is to have finitely many large jumps in finite time. Two regimes matter: if integral of min(1,|x|) nu(dx) < infinity the jumps are of finite variation and can be summed without compensation; otherwise only the compensated sum exists.
Reading the Lévy measure is how one identifies a process at a glance: nu = lambda delta_a is compound Poisson with fixed jump a; nu(dx) = c|x|^(-1-alpha) dx is alpha-stable; nu(dx) ~ c x^(-1) e^(-x) dx on x>0 is the gamma process. The standard misconception is that a finite Lévy measure is required — it is not; many of the most useful processes (stable, gamma, variance-gamma) have INFINITE Lévy measure, meaning infinitely many small jumps in every interval, and are nonetheless perfectly well-defined.
For the symmetric alpha-stable process, nu(dx) = c |x|^(-1-alpha) dx with 0 < alpha < 2. Integral of x^2 near zero behaves like integral of x^(1-alpha), which converges (since alpha < 2), so the small-jump condition holds; but integral of |x| near zero behaves like integral of x^(-alpha), which DIVERGES for alpha >= 1 — hence alpha-stable processes with alpha >= 1 have infinite-variation jumps and need compensation.
The x^(-1-alpha) singularity at 0 controls whether jumps are summable.
The condition is integral of min(1, x^2) nu(dx) < infinity, NOT nu(R\{0}) < infinity. A finite Lévy measure means compound Poisson (finitely many jumps per unit time); an infinite one means infinitely many small jumps, which is the generic case.