Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process X_t, as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t↓0; for example, we may have X_t/t →^p+∞ (t↓0) (weak divergence to +∞), whereas X_t/t→∞ a.s. (t↓0) is impossible (both are possible when t→∞), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. "Almost sure stability" of X_t, i.e., X_t tending to a nonzero constant a.s. as t→∞ or as t↓0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to "strong law" behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.