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.


The [Glivenko-Cantelli Theorem](http://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem) states that $\hat{F}\_n(x)$ converges to $F(x)$ uniformly, that is, $\sup\_{x\in\mathbb{R}}|\hat{F}\_n(x)-F(x)|\rightarrow 0$ almost surely as $n\rightarrow\infty$. The [Dvoretzky–Kiefer–Wolfowitz inequality](http://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality) quantifies the rate of convergence as follows: $P(\sup\_{x\in\mathbb{R}}|\hat{F}\_n(x)-F(x)|>\epsilon)\leq2e^{-2n\epsilon^2}$.


I am wondering if anything is known about the distributions of the errors in the empirical approximation of a CDF: that is, how is the random variable $\varepsilon\_n(x)\equiv\hat{F}\_n(x)-F(x)$ distributed? Is there perhaps asymptotic Gaussianity here, maybe for some classes of $F(x)$?


Here is the reason for my question: suppose I draw $nM$ random variables i.i.d. that have the common CDF $F(x)$ where $m=1,2,\ldots,M$, and compute $M$ empirical approximations as follows: 


$$\hat{F}\_{n,m}(x)=\frac{1}{n}\sum\_{i=n(m-1)+1}^{nm}\mathbf{1}\_{\{X\_i\leq x\}}(x)\tag{2}.$$


Now I need to approximate $F(x\_0)$ for a given $x\_0$. Which approach would yield a better estimate: the average $\frac{1}{M}\sum\_{m=1}^M\hat{F}\_{n,m}(x\_0)$ over the approximations in (2), or using $nM$ samples in constructing the approximation in (1)? If it helps, the distribution that I am trying to approximate is a mixture of binomial random variables.



What is known about the distribution of the errors in empirical approximation of a CDF?
I believe your two empirical approximations are the same: aren't they both the total fraction of observations less than or equal to $x\_0$?


As for the question of asymptotic normality, [Donsker's Theorem](http://en.wikipedia.org/wiki/Donsker%27s_theorem) gives the main result in this direction, namely that the processes $\sqrt{n}\left(\hat{F}\_n(x)-F(x)\right)$ converge in an appropriate sense to the process $B(F(x))$, where $B$ is a standard Brownian bridge.


