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

Let X = ( X ( t ): t ≥ 0) be a Lévy process and X ∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ 2 (∊) = var( X ∊ (1)). In simulation, X - X ∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X ∊ /σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X ∊ /σ(∊) 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.