Setup
Let $\mathbf X = (X_1, X_2, \dots)$ be a sequence of identically distributed random variables on a probability space $(\Omega, \mathcal F, P)$ with the common marginal distribution (function) $F$.
Set the empirical distribution function of $X_1, \dots, X_n$ as $\hat F_n: (\Omega, \mathbb R) \to [0,1]$, given by $$ F_n^\omega (t) := \frac 1n \sum_{j=1}^n \mathbf{1} \big\{ X_j(\omega) \leq t \big\}. $$
The Glivenko-Cantelli theorem tells us that if the $X_i$ are independent and identically distributed (i.i.d.), then $$ \| \hat F_n - F \|_\infty \overset{\text{def}}{=} \sup_{t \in \mathbb R} \Big | \hat F_n(t) - F(t) \Big| \overset{\text{a.s.}}{\longrightarrow} 0. $$
Question
Are there known weaker assumptions on $\mathbf X$ than independence of the $X_i$ under which $\hat F_n$ converges to $F$ uniformly in probability? That is: $$ \| \hat F_n - F \|_\infty \overset{P}{\longrightarrow} 0. $$
Note: I don't necessarily need a proof here. References would suffice.