We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate. A consequence of this MDP is the tightness of the method of bounded martingale differences in the regime of moderate deviations.
"Moderate Deviations for Martingales with Bounded Jumps." Electron. Commun. Probab. 1 11 - 17, 1996. https://doi.org/10.1214/ECP.v1-973