the Central Limit Theorem for Local Martingales states the following.
Theorem Let $M_n = (M_n(s))_{s \geq 0}$ be a square integrable local martingale such that for all $T > 0$ $$ \lim_{n \rightarrow \infty} \mathbb{E}[\sup_{0 \leq s \leq T}|\langle M_n\rangle_s - \langle M_n \rangle_{s-}|]=0$$ $$ \lim_{n \rightarrow \infty} \mathbb{E}[\sup_{0 \leq s \leq T}| M_n(s) - M_n(s-)|^2]=0$$ and for all $s>0$ for $ n\rightarrow \infty$ $$ \langle M_n\rangle_s \rightarrow^\mathbb{P} cs$$ for some c>0. Than it holds $$ M_n \rightarrow^d \sqrt{c}B.$$ for a Brownian Motion $B$.
My question: Is there some generalisation out there with a result like:
Let $M_n = (M(s))_{s \geq 0}$ be a square integrable local martingale such that for all $T > 0$ $$ \lim_{n \rightarrow \infty} \mathbb{E}[\sup_{0 \leq s \leq T}|\langle M_n\rangle_s - \langle M_n \rangle_{s-}|]=0$$ $$ \lim_{n \rightarrow \infty} \mathbb{E}[\sup_{0 \leq s \leq T}| M_n(s) - M_n(s-)|^2]=0$$ and for all $s>0$ for $ n\rightarrow \infty$ $$ \langle M_n\rangle_s \rightarrow^\mathbb{P} \int_0^s\eta(u)du$$ $$ M_n \rightarrow^d \int\sqrt{\eta}dB.$$ for a Brownian Motion $B$ and $\eta \in \mathcal{L}^2$ (possible under stronger requirements than stated above). According to my intuition it should at least hold for a continuous, monton growing functions $\eta \in \mathcal{C}$.