Let $X1 $, $X2 $, $X3 $...$Xn $ be n observations with distribution function $F $. Let $F^{*} $ be the empirical distribution of the random sample.
$F^{*} = \frac{1}{n} \sum I(X_{i} \le x)$ where I = 1, if the equality holds, 0 otherwise.
i) Find the distribution of $ \sum I(X_{i} \le x)$
ii) Find the limiting distribution of ${\sqrt(n)} [F^{*}(x)-F(x)]$
This is my approach;
Let r observations satisfy the equality and (n-r) observations don't.
$F^{*} = \frac{1}{n} (1+1+..1+0+0+..+0) = \frac{r}{n}$
$ \sum I(X_{i} \le x) = r$
So distribution is the number of successes in the sample.
But I'm not sure if this is the way to approach the problem.
For the 2nd part;
$ Y = {\sqrt(n)} [F^{*}(x)-F(x)] = {\sqrt(n)} [r/n - p] = \frac{1}{\sqrt(n)} [r-np]$
$ n \rightarrow \infty , r\rightarrow np$
therefore,
$ (r-np) \sim N(0, \sigma^{2})$
$ Y = \frac{1}{\sqrt(n)} [r-np] \sim N(0, \frac{\sigma^{2}}{n})$