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.

  • $\begingroup$ Corollary VIII 3.24 (a variant of the martingale convergence theorem) in the book by Jacod and Shiryaev might help, though I'm not entirely sure. $\endgroup$ – Mars Plastic Mar 19 at 17:15

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