(Hopefully last in a long series of posts from the "I don't have Rick Durrett's brain" department... apologies.)
In Durrett's proof of a a simple martingale CLT (Theorem 8.8.3, p. 341), he loses me in the second line of the proof. A new low, no doubt.
Theorem 8.8.3: Suppose $\{X_{n,m}, \mathcal{F}_{n,m}\}$ is a martingale difference array. If (i) for each $t$, $V_{n,[nt]} \to t$ in probability and (ii) $|X_{n,m}| \leq \epsilon_n$ for all $m$ with $\epsilon_n \to 0$, then $S_{n,(n\cdot)} \Rightarrow B(\cdot)$.
Proof: (i) implies $V_{n,n} \to 1$ in probability. By stopping each sequence at the first time $V_{n,k} > 2$ and setting the later $X_{n,m} = 0$, we can suppose without loss of generality that $V_{n,n} \leq 2+\epsilon_n^2$.
The quantity $V_{n,k}$ is defined as: $$ V_{n,k} = \sum_{1 \leq m \leq k} E(X_{n,k}^2 | \mathcal{F}_{n,m-1}) $$ I get that these $V$'s essentially represent the conditional variances of the martingale array, and I see how the assumptions imply that if we could perform the operation that Durrett suggests, then we can force it to be $\leq 2 + \epsilon_n^2$. But how can this operation (setting the $X_{n,m}$'s to 0) preserve generality?
I could see how rescaling the $X_{n,m}$'s by a common amount to reduce the conditional variances preserves generality, but I can't wrap my head around how to perform what he suggests without losing something. Can anyone provide a simple example to illustrate what is supposed to happen here?
The bound 2 is arbitrary in the sense that it can be replaced by any real number larger than 1 - the real requirement is that you set some bound on the summed conditional variances of the array. Note that you already assume that $V_{n,nt} \xrightarrow{P} t$ - what does this condition entail on the variances? One reason that this is allowed in general is that if I recall correctly, you can show that $\max E(X^2_{n,k}|\mathcal{F}_{n,m-1}) \xrightarrow{P} 0$ as $n \rightarrow \infty$ by showing $\mathcal{L}^1$ convergence.
I would have preferred this to have been a comment, but I do not have sufficient reputation for that - sorry.