Approximations of small jumps of Levy processes with a view towards simulation

Let X = (X(t) : t greater than or equal to 0) be a Levy process and X-epsil on the compensated sum of jumps not exceeding epsilon in absolute value, si gma (2)(epsilon) = var(X-epsilon(1)). In simulation, X - X-epsilon is easil y generated as the sum of a Brownian term and a compound Poisson one, and w e investigate here when X-epsilon / sigma (epsilon) can be approximated by another Brownian term. A necessary and sufficient condition in terms of sig ma (epsilon) is given, and it is shown that when the condition fails, the b ehaviour of X-epsilon / sigma (epsilon) can be quite intricate. This condit ion is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approxima tions.