I posted this question originally in math stack exchange, but I got no answer.

In wikipedia, there is a version of a CLT for Martingales, which I cannot find any reference to. ( https://en.wikipedia.org/wiki/Martingale_central_limit_theorem)

The theorem claims the following:

Let $X_1,X_2,...$ be a martingale with bounded increments, i.e.

$[\mathbb{E} [ X_{t+1}-X_t | X_1,...,X_t]=0$ and $|X_{t+1}-X_t|\le k$ almost surely for some $k$ and all $t$.

Define $\sigma_{t}^2=\mathbb{E}[(X_{t+1}-X_t)^2|X_1,...,X_t]$, and let $\tau_\nu=min\{t\ :\ \Sigma_{i=1}^{t} \sigma_i^2 \ge \nu$ }.

Then $\frac{X_{\tau_{\nu}}}{\sqrt{\nu}}$ converges to $N(0,1)$ in distribution as $\nu \longrightarrow \infty$.

I would like to know how to prove this or if there is any reference on the web.

Thanks!