Necessary and sufficient condition for $\cM_2$-convergence to a Lévy process for billiards with cusps at flat points

Abstract

We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding Lévy process in the $\cM_1$ and $\cM_2$ Skorohod topologies has been studied in recent work. Here we show that certain sufficient conditions for $\cM_2$-convergence are also necessary.

Updated to take into account finalized versions of refs [2] and [4]