Almost sure uniform convergence of the empirical distribution

Question

Let $U_1, U_2, \dots \overset{\text{iid}}{\sim} U[0,1]$, i.e. a sequence of independent and identically distributed uniform random variables on the set $[0,1]$.

By the Strong Law of Large Numbers, we know that for any $t \in [0,1]$ $$ \left|\frac{1}{n} \sum_{j=1}^n \mathbf{1}\{U_j \leq t\} - t\right| \to 0 \quad \text{almost surely}, $$ since $\mathbb{E}(\mathbf{1}\{U_j \leq t\}) = \mathbb{P}(U_j \leq t) = t$. Thus, we have pointwise almost sure convergence.

How can we then show that uniform almost sure convergence also holds? That is $$ \sup_{0 \leq t \leq 1}\left|\frac{1}{n} \sum_{j=1}^n \mathbf{1}\{U_j \leq t\} - t\right| \to 0 \quad \text{almost surely}. $$

Answer 1

I realized after writing this question, that this is just one case of the Glivenko-Cantelli Theorem:

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of for real-valued independent and identically distributed random variables. Let $F$ be the distribution function of $X_1$ and define the empirical distribution function by $\hat{F}_n(t) := \frac{1}{n} \sum_{j=1}^n \mathbf{1}\{X_j \leq t\}$. Then $$ \|\hat{F}_n - F\|_\infty \to 0 \quad \text{almost surely}. $$ Since it's a well-known theorem, proofs are not difficult to find.

Answer 2

The following result is found in many books, in particular K L Chung's book:

If $F_n,F$ are distribution functions such that $F_n(t) \to F(t)$ on dense set of points $t$ and if $F$ is continuous then $F_n \to F$ uniformly.

In our case there is a null set outside which $F_n(t) \to F(t)$ for all $t$ rational. Hence the result follows.