CLT for Martingales

Question

I posted this question originally in math stack exchange, but I got no answer. (https://math.stackexchange.com/questions/2604591/clt-for-martingales)

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 $\tau_\nu=\min\left\{t\ :\ \sum_{i=1}^{t} \sigma_i^2 \ge \nu\right\}$.

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!

Answer 1

It is not exactly the mentioned result, but in

Ouchti, Lahcen On the rate of convergence in the central limit theorem for martingale difference sequences, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 1, 35–43.

(here is a link where this paper has the availability it should have)

there is a related result, in the following sense:

• one needs the assumption that $\sum_t\sigma^2_t$ is almost surely infinite (which is a reasonable assumption).
• The condition of boundedness of the increments is replaced by the weaker assumption of the existence of a sequence of random variables $\left(Y_t\right)_{t\geqslant 1}$ having a finite fourth order moment and such that $Y_t\geqslant 1$ almost surely for all $t$ and $\mathbb E\left[ \left\lvert X_t\right\rvert^3\mid\mathcal F_{t-1}\right]\leqslant Z_t\sigma_{t-1}^2$.
• A convergence rate is given, in term of the uniform norm of the cumulative distribution function of $X_{\tau_n}/\sqrt n$ with that of a standard normal random variable. In the bounded case, the rate is of order $n^{-1/4}$.

Answer 2

This Theorem is a direct consequence of the Skorohod representation of Martingales. You can find it, along with many variants, in Hall, Peter, and Christopher C. Heyde. Martingale limit theory and its application. Academic press, 2014.