small versus large jumps
Splitting a Lévy process's jumps into small ones and large ones is the technical heart of the whole construction. The two kinds behave completely differently — large jumps are rare and integrable, small jumps may be infinitely numerous and must be tamed by compensation — and getting the split right is what makes the Lévy-Itô decomposition converge.
Fix a threshold, conventionally 1. Large jumps are those with |Delta X| >= 1: by the integrability of nu away from zero, they occur at finite rate nu(|x| >= 1) < infinity, so in any finite interval there are only finitely many. They form a compound Poisson process and can be added to the path directly — no centering is needed, and indeed centering could be impossible if nu has heavy tails so that the large jumps lack a mean. Small jumps are those with |Delta X| < 1: their rate nu(0 < |x| < 1) can be infinite, so they may arrive infinitely often, but the constraint integral over (|x|<1) of x^2 nu(dx) < infinity guarantees their compensated sum integral of x(N(t,dx) - t nu(dx)) converges in L^2 as a martingale. Subtracting the deterministic compensator t nu(dx) is what kills the otherwise-divergent drift produced by accumulating tiny jumps.
This dichotomy is why the Lévy-Khintchine integrand carries the indicator 1_{|x|<1} only on the linear term: small jumps need their first moment subtracted, large jumps do not. Operationally it drives simulation (sample finitely many large jumps exactly, approximate the small-jump martingale by a Brownian-motion proxy of matching variance, the small-jump CLT) and analysis (the small-jump part is L^2-bounded, the large-jump part is a clean compound Poisson). The honest caveat: the threshold is arbitrary, and a quantity that looks like a drift may simply be the chosen compensation — so a Lévy process's drift parameter is meaningful only once the cutoff is fixed.
Take an alpha-stable process with 1 < alpha < 2. The large jumps (|x| >= 1) arrive at finite rate and lack a finite mean when alpha < 1 but here alpha > 1 so they are integrable; the small jumps arrive infinitely often (nu(|x|<1) = infinity) and only their compensated sum converges. Simulating it: draw finitely many large jumps from the truncated stable measure, and replace the small-jump martingale by a Gaussian of variance integral over (|x|<eps) of x^2 nu(dx).
Add large jumps raw; compensate the infinitely many small jumps.
The 1 in |x| >= 1 is a free convention; any positive cutoff works and only relocates the drift. Large jumps are never compensated (their compensator may diverge); small jumps must be (their raw sum may diverge).