central limit theorem for continuous-time martingales

Question

I'm searching for a proof of the following result: If $(M_t)_{t\ge0}$ is a square-integrable martingale with statinary increments such that $$\frac1t[M]_t\xrightarrow{t\to\infty}\sigma^2$$ in $L^1$ for some $\sigma>0$, then $$\frac{M_t}{\sqrt t}\xrightarrow{t\to\infty}\mathcal N(0,\sigma^2)$$ in distribution.

I wasn't able to find this result in any textbook which I've consulted.

Answer 1

What you usually do in this kind of contexts is to deduce the continuous-time martingale theorem from the discrete-time martingale analogue theorem.

There is a paper showing that kind of method :

Central Limit Theorems for Martingales with Discrete or Continuous Time Inge S. Helland

Scandinavian Journal of Statistics Vol. 9, No. 2 (1982), pp. 79-94 (16 pages)