Small-time moment asymptotics for Lévy processes

Conditions ensuring that limt→01tEf(Xt)=∫f(x)ν(dx) are given for a Lévy process XX with Lévy measure νν and for unbounded moment functions ff. Compared with previous works, the moment functions considered here satisfy very mild conditions aimed at controlling how fast ff grows at infinity. As an application of our results, the infinitesimal generator of the Lévy process is shown to be well-defined in a class of smooth unbounded functions equipped with a suitable norm. Also, the rate of convergence is studied when ff is a smooth function vanishing in a neighborhood of the origin.