For i.i.d. observations, the Dvoretzky–Kiefer–Wolfowitz inequality tells us that the empirical c.d.f. almost surely converges to the population c.d.f. (i.e., $\underset{x \in \mathbb{R}}{\sup} | F_n(x) - F(x)| \to 0$ almost surely), but I wonder whether such a result holds when the observations are drawn from a mixing ($\alpha$-mixing or $\beta$-mixing) stochastic process?
For the intuition, I believe that compared with drawing i.i.d. examples, drawing from a dependent stochastic process does not change the fact that for any $\epsilon > 0$, $\mathbb{P} \left[\underset{x \in \mathbb{R}}{\sup} | F_n(x) - F(x)| > \epsilon \right] \to 0$ as $n \to \infty$. Perhaps the convergence is slower than the case with i.i.d. observations, while I believe it converges to zero.
However, I have searched a lot on the internet but do not find such a result.
