Central Limit Theorem for Multinomial Trials to prove weak convergence to Brownian Bridge I'm trying to demonstrate that if we define the empirical process by
$X_t^n=\sqrt n (F_n(t)-t)$,
where $$F_n(t)=\frac{1}{n}\sum_{i=1}^nI_{\xi_i\leq t},$$
and $\xi_i$ are independent uniform random variables over (0,1), then the finite-dimensional distributions of the $X^n$ converge weakly to those of $W^º$, the brownian bridge on (0,1).
I was trying to follow Billingsley's proof (Convergence of Probability Measures 2ed. pp149), but he mentioned that it follows by the Central Limit Theorem for Multinomial trials. But I ended up proving it by the Multivariate Central Limit Theorem, as I explain it below.

Just as A.S. commented, if we denote by $Z_i=1_{\xi_i\leq t_i}-t_i$ for some $k$ different times $t_i\in (0,1)$ then the collection $\lbrace Z_i \rbrace_{i=1}^k$ are independent and identically distributed random vectors in $\mathbb{R}^n$ with mean $\overline{0}$ and $\Sigma$ VarCov matriz with elements $\lbrace a_{ij} \rbrace$, $a_{ij}=\min (t_i,t_j)-t_it_j$, which determines the finite dimensional distributions of the brownian bridge.
