I am studying Lecture 1 of Stefan Wager's Causal Inference notes and come across a central limit theorem for the difference-in-means estimator, which I am unable to prove.
The mathematical abstraction is as follows. Let $W_1,...,W_n$ be i.i.d. Bernoulli random variables. Let $(Y_1(0),Y_1(1)),...,(Y_n(0),Y_n(1))$ be i.i.d. random vectors. Moreover, assume that $\{W_i\}_{i=1}^n$ and $\{(Y_i(0),Y_i(1))\}_{i=1}^n$ are independent. Let $n_1 = \sum_{i=1}^n W_i$ and $n_0=n-n_1$, which are both random variables. The difference-in-means estimator is $$ \widehat{\tau} = \frac{1}{n_1}\sum_{i=1}^nW_iY_i - \frac{1}{n_0}\sum_{i=1}^n(1-W_i)Y_i. $$ Define $\tau =\mathbb{E}[Y_1(1)-Y_1(0)]$. On the top of page 5, the author states the following:
A standard central limit theorem can be used to verify that $$ \begin{align} & \sqrt{n}(\widehat{\tau}-\tau)\Rightarrow\mathcal{N}(0,V), \\[4pt] & V = \text{var}(Y_i(0)) / \mathbb{P}(W_i=0) + \text{var}(Y_i(1)) / \mathbb{P}(W_i=1). \end{align} $$
My question: How to prove this rigorously?
My thoughts: Some useful properties of $\widehat{\tau}$ are $$ \begin{align} & \mathbb{E}\left[\widehat{\tau}\mid\{(Y_i(0),Y_i(1))\}_{i=1}^n,n_1\right] = \frac{1}{n}\sum_{i=1}^n(Y_i(1)-Y_i(0)), \\[4pt] &\text{var}(\widehat{\tau}\mid n_1) = \frac{1}{n_0}\text{var}(Y_i(0)) + \frac{1}{n_1}\text{var}(Y_i(1)). \end{align} $$ These two results are stated in pages 3 and 4 in the notes, and I am able to prove them rigorously. By the first equality, $\mathbb{E}\widehat{\tau}=\tau$. Intuitively, for $n$ large, $\sqrt{n}(\widehat{\tau}-\tau)$ approximates a zero-mean normal random variable with variance $$ (\sqrt{n})^2\text{var}(\widehat{\tau}\mid n_1) = \frac{\text{var}(Y_i(0))}{n_0/n} + \frac{\text{var}(Y_i(1))}{n_1/n} \to \frac{\text{var}(Y_i(0))}{\mathbb{P}(W_i=0)} + \frac{\text{var}(Y_i(1))}{\mathbb{P}(W_i=1)}. $$ However, the standard central limit theorem does not really apply because we are adding dependent random variables $(W_n/n_1-(1-W_i)/n_0)Y_i$, $1\le i\le n$.
Alternatively, I could apply central limit theorem to $\frac{1}{n}\sum_{i=1}^nW_iY_i$ and the strong law of large numbers to $n_1/n$, and use Slutsky's Theorem to combine them, then do the same for $\frac{1}{n}\sum_{i=1}^n(1-W_i)Y_i$ and $n_0/n$. But then it is not clear how to deduce the weak limit of their difference $\widehat{\tau}$.