Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly increasing.
Show that for $Z_{n}:=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}(1+\log{(1-F_{k}(X_{k})))}$
$Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$
My ideas: I've realized there are two ways to show convergence in distribution, namely
through the
$i)$ Central Limit Theorem
$ii)$ or the very definition of the Cumulative distribution function
I suspect in this case, given the definition of $Z_{n}$, the CLT will be useful
It is clear since $(F_{k})_{k \in \mathbb N}$ are continuous that $(F_{k}(X_{k}))_{k \in \mathbb N}$ are independent random variables
The only problem is I see no way to get $(1+\log{(1-F_{k}(X_{k})))}$ into the form $\frac{F_{k}(X_{k})-\mathbb E [F_{k}(X_{k})]}{\sigma}$ that is necessary for the CLT.
Any ideas?
Hints: Let $U_k=F_k(X_k)$. Verify that $U_k$ 's are i.i.d with uniform distribution on $(0,1)$. Apply standard form of CLT.