Compactness and continuity properties for a lévy process at a two-sided exit time

We consider a Lévy process X = (X(t))_t≥0 in a generalised Feller class at 0, and study the exit position, |X(T (r))|, as X leaves, and the position, |X(T (r)−)|, just prior to its leaving, at time T (r), a two-sided region with boundaries at ±r, r > 0. Conditions are known for X to be in the Feller class F C₀ at zero, by which we mean that each sequence t_k ↓ 0 contains a subsequence through which X(t_k), after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on X to characterise similar properties for the normed positions |X(T (r))| /r and |X(T (r)−)| /r, and also for the normed jump |∆X(T (r))/r| = |X(T (r)) − X(T (r)−)| /r (“the jump causing ruin"), as convergence takes place through sequences r_k ↓ 0. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.