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-epsilon the compensated sum of jumps not exceeding epsilon in absolute value, sigma (2)(epsilon) = var(X-epsilon(1)). In simulation, X - X-epsilon is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X-epsilon / sigma (epsilon) can be approximated by another Brownian term. A necessary and sufficient condition in terms of sigma (epsilon) is given, and it is shown that when the condition fails, the behaviour of X-epsilon / sigma (epsilon) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.