# Convergence in distribution with empirical distribution function (EDF)

I'm struggling with the following question:

Let $$F_n(x)$$ denote the EDF of a random sample. Show that

$$\sqrt{n}(F_n(x)-F(x))\xrightarrow[]{d}N(0,F(x)(1-F(x)))$$.

I think that the course of action should be:

1. prove that $$F_n(x) \xrightarrow[]{p} F(x)$$ using the WLLN and
2. prove the convergence in the question using the CLT.

This is my take on the first step: $$F_n(x)=\frac{1}{n} \sum_{i=1}^n \mathbf{1}(x_i \leq x)$$ where $$\mathbf{1}(\cdot)$$ is the indicator function. Then, for all $$x$$, $$\mathbf{1}(x_i \leq x)$$ is an iid random variable with expectation $$F(x)$$. Thus, by the WLLN, $$F_n(x)$$ is a consistent estimator of $$F(x)$$: $$F_n(x) \xrightarrow[]{p} F(x)$$.

So far, so good?

My real troubles begin with the second step. I have set up the expression

$$\sqrt{n}(F_n(x)-F(x)) = \sqrt{n}\left(\frac{1}{n} \sum_{i=1}^n \mathbf{1}(x_i \leq x) - F(x)\right)$$

but I don't really know where to go from there. Any pointers would be greatly appreciated!

• What do you know? what did you try? Feb 10, 2015 at 15:37
• Thus far, very little. I did not really know where to start. Apparently, F_n(x) converge in distribution to F(x) by WLLN and the expression in the question should then be derived from the CLT. Feb 10, 2015 at 16:49
• Fix $x$. Then think about variables $Z_i=1(x_i\le x)$. Are they independent? Are they identically distributed? What is their mean and variance? Feb 11, 2015 at 16:54
• Note if $x$ is not fixed, then the question becomes much harder. Feb 11, 2015 at 16:55
• Thanks, @mpiktas ! If you post those two comments as an answer, then I can mark the question as answered. And, a final question, does this mean that my first step is not really needed to answer the question? Feb 12, 2015 at 8:36

Fix $x$. Then think about variables $Z_i=1(x_i\le x)$. Are they independent? Are they identically distributed? What is their mean and variance?
Note that if $x$ is not fixed, then the question becomes much harder.