Let $(M_t)_{t \in \mathbb{R}^*}$ be a square-integrable martingale. I am looking for a reference for the following convergence result : $$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$ as $t \to \infty$, $\xi \sim \mathcal{N}(0, 1)$ and where $\langle M_t \rangle = E(M_t^2)$ is the square-braket of a stochastic process.
The result seems clear if we sample the martingale at the integer values. In fact, we can define, for $n, k \in \mathbb{N}^*$, $$X_{n, k} = \frac{M_{k} - M_{k - 1}}{\sqrt{E(M_n^2)}}$$ which is a martingale difference sequence and then the standard CLT's yield the result. But what about the case when $t$ is a continuous parameter ? Does anybody have a reference ? Thanks a lot.