I'd like to know if there exists a martingale central limit theorem of a particular type.
Theorem (?) Suppose that $\{(\xi_{n,k}, \mathcal{F}_{n,k})\}$ are a martingale difference array that satisfies
$$ \frac{\max_{1 \leq k \leq n} |\xi_{n,k}|^2}{\sum_{k=1}^{n} \xi_{n,k}^2} \xrightarrow{p}{0} \quad \text{as} \quad n \to \infty$$
Then the following self-normalized random variable is asymptotically normally distributed:
$$\frac{\sum_{k=1}^{n} \xi_{n,k}}{\sqrt{\sum_{k=1}^{n} \xi_{n,k}^2}} \xrightarrow{d} \mathcal{N}(0,1)$$
Question: Is this theorem true? If not, what other conditions must be satisfied?