Investigating weak topologies for probability measures

My early research was focused on weak topologies of probability measures associated to stochastic processes, via Martingale Theory and the General Theory of Processes. The first article I published in this direction of research was [3] devoted to the convergence in distribution of continuous martingales. This was followed and improved by a series ([4], [5], [6]) of notes which included a first version of a Central Limit Theorem for (discontinuous) square integrable Martingales, building up a new method to approach the convergence in distribution of stochastic processes. The first applications of this method appeared in [7] and [8]. One of my main papers on the method I built up is the memoir [9]. Furthermore, several extensions and improvements of the method followed in [10]-[14]. The article [15] provided a complete description of tightness on the Skorokhod space D by means of stopping times. On the other hand, in [16] the more general result on the Central Limit Theorem for Local Martingales was established. My research turned then into the search of necessary and sufficient conditions for the validity of the Central Limit Theorem. This was achieved for the case of semimartingales in [17]. Semimartingale or martingale problems have been considered in [18] to [22] and [26]. In particular, various results on the approximation of diffusions were derived. Finally, in [23], a paper written jointly with Eckhard Platen, we analysed discretisation procedures and the approximation of diffusions. The study of metastable phenomena in stochastic particle models, motivated the search of a weaker topology on the space D replacing the customary Skorokhod's topology. The articles [24], [25], [27], [28] were aimed at solving that problem. Finally, the convergence of non adapted processes was covered in [29], [35], [36], and more recently, random fields convergence was studied in [37].