The stable central limit theorem for local martingales with bounded jumps via Skorohod embedding

The stable central limit theorem for properly normalized local martingales with bounded jumps is proved. Instead of the usual characteristic function-type methods we use an embedding technique in combination with a result on nested Brownian motions. In this approach, the stability of the CLT is explained by the fact that nested Brownian motions are asymptotically independent of any other random element. As was previously shown in the special case of continuous local martingales, the embedding technique leads to short and transparent arguments. In the conclusion we discuss the direction in which further research is needed to make the embedding method applicable in an even larger number of situations.