CLT for martingales-III: discontinuous compensators

TL;DR

This paper introduces a new weak convergence theorem for martingales with discontinuous compensators, relaxing traditional conditions and utilizing advanced topologies, with applications including Brownian subordinators.

Contribution

It presents a novel weak convergence result for martingales with discontinuous compensators, expanding the theoretical framework beyond classical quadratic variation convergence.

Findings

Established a new weak convergence theorem for martingales with discontinuous compensators.

Demonstrated the theorem's applicability to Brownian subordinators.

Provided alternative conditions for discrete-time martingale differences and transforms.

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.