If we have the following estimator: $\hat{F_Z}(z)=\frac{1}{N}\sum_{i=1}^N1\{Z_i\leq z\}$. The CDF of $Z$ is defined as $F_Z(z)=Pr(Z\leq z)$. $Z_1, ..., Z_N$ is i.i.d. data.

What would be the steps to show that $\hat{F}$ is consistent and asymptotically normal and to find the asymptotic variance at a given point $z$.

My thought was to get the pdf but apparently $\hat{F}$ is not very useful for estimating the PDF. That I don't understand why?

I am not sure but this is what I got so far: $$\hat{F_Z}(z)=\frac{1}{N}\sum_{i=1}^N1\{Z_i\leq z\}\xrightarrow{LLN}E[1(Z_i\leq z)]=Pr(Z\leq z)=F_Z(z)$$

$$\sqrt{N}(N^{-1}\sum 1(Z_i\leq z)-E[1(Z_i\leq z))\xrightarrow{CLT}N(0, Var(1(Z\leq z)))$$

The questions are related so I merged them together but if needed I can post a new one.

If we would consider now a random variable U independent of Z with CDF $F_U(\cdot)$, and a symmetric PDF, and consider some $h > 0$. We now consider a different estimator $$\tilde{F}(z)=\frac{1}{N}\sum_{i=1}^NF_U[\frac{z-Z_i}{h}]$$ Is $\tilde{F}$ consistent for $F_Z$?

For this, I guess we would be using the nonparametric approach. Would we follow the same steps as with the previous one or is it a different thing?