Helland (1982) (Theorem 2.5) gives the following conditions for a martingale central limit theorem.
Given a triangular martingale difference array $\{(\xi_{n,k}, \mathcal{F}_{n,k})\}$, if any of the following sets of conditions below is satisfied, then a martingale CLT holds:
$$\sum_{k=1}^{n} \xi_{n,k} \xrightarrow{d} \mathcal{N}(0,1)$$
Set 1
1.a) $\sum_{k=1}^{n} E[\xi_{n,k}^2 I\{|\xi_{n,k}| > \epsilon \} | \mathcal{F}_{n,k-1}] \xrightarrow{p} 0\quad$ for all $\epsilon > 0$
1.b) $\sum_{k=1}^{n} Var[\xi_{n,k} | \mathcal{F}_{n,k-1}] \xrightarrow{p} 1$
Set 2
2.a) $\sum_{k=1}^{n} \xi_{n,k}^2 \xrightarrow{p} 1$
2.b) $E[\max_{1 \leq k \leq n} |\xi_{k,n}|] \to 0$
Set 3
3.a) $\max_{1 \leq k \leq n} |\xi_{k,n}| \xrightarrow{p} 0$
3.b) $\sum_{k=1}^{n} \xi_{n,k}^2 \xrightarrow{p} 1$
3.c) $\sum_{k=1}^{n} \left|E[\xi_{n,k}I\{|\xi_{n,k}| > 1\} | \mathcal{F}_{n,k-1}] \right| \to 0$
Question: What kind of assumptions imply the conditional variance / quadratic variability conditions (1.b), (2.b) or (2.c)?
For example, Hall and Heyde (1980, p53) comment that martingale arrays are commonly constructed from ordinary martingale differences $\{ (X_{k}, \mathcal{F}_{k}) \}$ by
$$\xi_{n,k} := \frac{X_{k}}{\sqrt{\sum_{k} E[X_{k}^2]}}$$
In this case, e.g. condition 1.b implies that the sum of conditional variances increases at exactly the same rate as the unconditional sum:
$$\frac{\sum_{k} E[X_{k}^2| \mathcal{F}_{k-1}]}{\sum_{k} E[X_{k}^2]} \xrightarrow{p} 1$$
When is this sort of condition true?