Central Limit Theorem with two iid

Question

Suppose$\left \{ X_1,\dots,X_n \right \}$ is a sequence of independent and identically distributed(i.i.d) random variables with finite expected value $E[X_i]=\mu _{x}$ and variance $Var[X_i]=\sigma_x^{2}$. And $\left \{ \alpha _1,\dots,\alpha _n \right \}$ is another sequence of i.i.d random variables with finite expected value $E[X_i]=\mu _{\alpha }$ and variance $Var[X_i]=\sigma_\alpha ^{2}$.Then, as $n$ approaches infinity, can we prove the random variables $\frac{1}{n} \sum_{i=1}^{n} \alpha _iX_i$ converge in to a normal distribution ${\mathcal {N}}(u,\sigma ^{2})$.

Answer 1

Your normalization is wrong. [For eample, $\frac{1}{n} \sum_{i=1}^{n}X_i$ itself does not converge!].

If $(\alpha_i)$ is independent of $(X_i)$ then $(\alpha_iX_i)$ is also i.i.d. with finite variance and standard CLT applies. Otherwise, you cannot assert anything about $ \sum_{i=1}^{n} \alpha _iX_i$