Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

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