I just learned martingale central limit theorem and got a problem at hand and do not know how to form the correct martingale.
Suppose we draw balls successively from a box of $2n$ balls and $n$ balls are marked with -1 and the other half are marked by 1.
Now, label these successive draws $X_1,X_2,\dots,X_{2n}$, and for each $m\leq 2n$,write $S_{m,2n} = \sum_{i=1}^{m} X_i$. Try to prove that $\forall t\in(0,1)$, $$\frac{S_{[2nt],2n}}{\sqrt{n}} \rightarrow ^D N(0,v_t^2).$$
I was thingking that maybe $S_{m,2n}$ would be some form of martingale but it seems not so and I do not know what to do next. Any idea on forming the martingale?