On the limiting behaviour of Lévy processes at zero (Q2464670)

This paper studies the limiting behavior of Lévy processes as time tends to zero; the author focuses on pure jump Lévy processes that are not of compound Poisson type. A central result characterizes the probability that a Lévy processes leaves a symmetric interval upwards and states, among other results, the following equivalence: \[ P(X_{T_r}>0) \rightarrow 1 \Leftrightarrow P(X_{t}>0) \rightarrow 1 \] as \(r,t\rightarrow0\), where \(X\) is a Lévy process and \(T_r=\inf\{s\geq0;| X_s| >r\}\).