Im interested in higher order inference (like Edgeworth expansions for iid case) of martingales, and reading series of paper by Per Mykland (1993, 1995, etc). In it, the core condition for the higher order expansion is the CLT(Central Limit Theorem) for the (predictable) quadratic variation of the martingale.
So, simply put, can anyone come up with a good way to prove $ n^{1/2}(\langle X_n \rangle - E[ \langle X_n \rangle ]) $ converges to a normal random variable, without assuming a lot more than the conditions necessary for proving the CLT for $ X_n $?