Let $X\_1,X\_2,\ldots,X\_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:


$$\hat{F}\_n(x)=\frac{1}{n}\sum\_{i=1}^n\mathbf{1}\_{\{X\_i\leq x\}}(x)\tag{1}$$


where $\mathbf{1}\_A(x)$ is an indicator function: unity when $x\in A$ and zero otherwise.


Per an answer to my [previous question](https://mathoverflow.net/questions/161222/what-is-known-about-the-distribution-of-the-errors-in-empirical-approximation-of), by [Donsker's Theorem](http://en.wikipedia.org/wiki/Donsker%27s_theorem) the scaled error process $G\_n(x)\equiv\sqrt{n}\left(\hat{F}\_n(x)-F(x)\right)$ converges to the Brownian bridge process $B(F(x))$.


I am interested in the error of a point estimate $\hat{F}\_n(x\_0)$. The first sentence of the "History" section of the [Wikipedia article on the Donsker's Theorem](http://en.wikipedia.org/wiki/Donsker%27s_theorem) states the following:



> 
> By the classical central limit theorem, for fixed $x$, the random variable $G\_n(x)$ converges in distribution to a Gaussian (normal) random variable $G(x)$ with zero mean and variance $F(x)(1 − F(x))$ as the sample size $n$ grows.
> 
> 
> 


I have two questions about this statement (in order of importance):


1. What is the standard approach when $F(x)$ is unknown? Can one use the estimate $\hat{F}\_n(x)$? (similarly to how
we use the sample variance estimator to approximate the confidence
interval for unknown mean given unknown variance)
2. While I am familiar with the standard proofs and the use of the
classical CLT, I don't see how one proves the statement above using standard CLT. Can
someone provide a hint?



Estimating the variance of error in empirical approximation to a distribution
If you're only interested in estimating $F$ at a single point $x\_0$, then you are really just estimating the probability $p\_0 = \mathbb{P}\left(X\in (-\infty,x\_0]\right)$. The indicator variables $B\_i = [X\_i\leq x\_0]$ are i.i.d. Bernoulli with (potentially unknown) probability $p\_0$, so you can apply the usual CLT machinery to get the stated result and from there a confidence interval for $p\_0$.


Similar methods work if you care about $F$ at finitely many points. You only really need Donsker's Theorem to understand the whole process $G$ at once.


