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We prove convergence to a Levy process for a class of dispersing billiardswith cusps. For such examples, convergence to a stable law was proved by Jung &Zhang. For the corresponding functional limit law, convergence is not possiblein the usual Skorohod J_1 topology. Our main results yield elementary geometricconditions for convergence (i) in M_1, (ii) in M_2 but not M_1.  In general, we show for a large class of nonuniformly hyperbolic systems howto deduce functional limit laws once convergence to the corresponding stablelaw is known.

Convergence to a L\'evy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps