CLT for martingales-III: discontinuous compensators

Abstract

We propose a new weak convergence theorem for martingales, under gentler conditions than the usual convergence in probability of the sequence of associated quadratic variations. Its proof requires the combined use of Skorohod's $\cJ_1$-topology and $\cM_1$-topology on the space of c\`adl\`ag trajectories. The emphasis is on those instances where the sequence of martingales or its limit is a mixture of stochastic processes with discontinuities. Alternative conditions are set forth in the special cases of arrays of discrete time martingale differences and martingale transforms. Examples of applications are provided, notably when the limiting process is a Brownian subordinator.