I'm having trouble with this question.
Let $X_1, \ldots, X_n$ be a random sample from some distribution with mean $\mu_1$ and variance $\sigma_1^2>0$. Let $Y_1, \ldots, Y_n$ be a random sample from some other completely unrelated distribution with mean $\mu_2$ and variance $\sigma_2^2>0$. Prove that as $n \to \infty$, $$ Z_n = \frac{(\bar{X}_n - \bar{Y}_n) - (\mu_1-\mu_2)} {\sqrt{\left(\sigma_1^2 + \sigma_2^2 \right)}/n} \to \mathcal{N}(0,1). $$
The fact that I don't know the distribution of $X_i$ and $Y_i$ throws me off. Does the CLT always go to $\mathcal{N}(0,1)$?
If you define $A_n = X_n - Y_n$ then you don't know the distribution of $A_n$ but its mean is $\mu_A = \mu_1 - \mu_2$ and its variance is $\sigma_A^2 = \sigma_1^2 + \sigma_2^2$ since $X$ and $Y$ are unrelated (i.e. independent) processes.
Hence, $$ Z_n = \frac{\frac{1}{n}\sum_{k=1}^n A_n - \mu_A}{\sigma_A/n} $$
and you can apply the CLT.