Suppose I have a sample $X_1, dots, X_n$, drawn from some distribution $mathcal{P}$ with cdf $F_X(x)$. Empirical distribution function is calculated as
$$ hat{F}(x) = frac{1}{n} sum^n_{i=1} I(X_i < x).$$
From the probabilistic point of view, for each $x$, $hat{F}(x)$ is a random variable (since it is the sum of random variables, since sample is viewed as iid random variables).

Also from theory I know that [probability integral transform]:
$$F_X(X) sim U(0, 1).$$

Now, from the “statistics” point of view $hat{F}(X_i)$ makes sense, as it is just some function calculated at some point, however from “probabilistic” point of view I have no clue what is $hat{F}(X_i)$, since it looks like a composition of random variables…

Now my question has two parts:

1. Is there any theoretical way to justify what is $hat{F}(X_i)$?
2. I want to find some sort of convergence of $hat{F}(X_i) to U(0, 1)$ as $n to infty$. I know that $hat{F}(x) stackrel{a.s.}{to} F(x)$ for each $x$, but how could it help…