Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Let for each n is an element of N X-n be an R-d-valued locally square integrable martingale w. r. t. a filtration (F-n (t), t is an element of R+) (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 -> infinity and the relation E(f(< zX(n)>(t))|F-n (s)) - f(< zX(n)>(t)) (sic) 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 X-n's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes (X-n(0), < X-n >) converge in distribution to some (X, H), then a sequence (X-n) converges in distribution to a continuous local martingale X with initial value (sic) and quadratic characteristic H, whose finite-dimensional distributions are explicitly expressed via those of (X, H).