Convergence of locally square integrable martingales to a continuous local martingale (Q764412)

Summary: Let, for each \(n \in \mathbb N\), \(X_n\) be an \(\mathbb R^d\)-valued locally square integrable martingale w.r.t. a filtration \((\mathcal F_n(t), t \in \mathbb R_+)\) (the probability spaces may be different for different \(n\)). It is assumed that the discontinuities of \(X_n\) are in a sense asymptotically small as \(n \rightarrow \infty\) and the relation \(\text{E}(f(\langle zX_n \rangle(t))|\mathcal F_n(s)) - f(\langle zX_n \rangle(t)) @> \text{P} >> 0\) holds for all \(t > s > 0\), row vectors \(z\) and bounded uniformly continuous functions \(f\). Under these two principal assumptions and a number of technical ones, it is proved that the martingales \(X_n\) are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes \((X_n(0), \langle X_n \rangle)\) converge in distribution to some \((\overset \circ X, H)\), then a sequence \((X_n)\) converges in distribution to a continuous local martingale \(X\) with initial value \(\overset \circ X\) and quadratic characteristic \(H\), whose finite-dimensional distributions are explicitly expressed via those of \((\overset \circ X, H)\).