Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Let for each ∈ℕ be an ℝ-valued locally square integrable martingale w.r.t. a filtration (ℱ(),∈ℝ+) (probability spaces may be different for different ). It is assumed that the discontinuities of are in a sense asymptotically small as →∞ and the relation ((⟨⟩())|ℱ())−(⟨⟩())→0 holds for all >>0, row vectors , and bounded uniformly continuous functions . Under these two principal assumptions and a number of technical ones, it is proved that the 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes ((0),⟨⟩) converge in distribution to some (∘,), then a sequence () converges in distribution to a continuous local martingale with initial value ∘ and quadratic characteristic , whose finite-dimensional distributions are explicitly expressed via those of (∘,)