The Dvoretzky–Kiefer–Wolfowitz inequality states the following:
Let $X_1,X_2,X_3,\ldots$ be i.i.d. random variables with cumulative distribution function $F$ and let $$F_n(x)= \frac{1}{n} \sum_{i=1}^n\pmb{1}_{\{X_i\leq x\}}$$ be the empirical distribution function, then $$ P\left( \sup_{x\in \mathbb{R}} |F_n(x) -F(x)| \geq \epsilon\right)\leq 2e^{-2n\epsilon^2}.$$
My question: are there similar bounds in case the $X_i$ are not identically distributed (but still independent) and $X_i \xrightarrow{D} X$ for some random variable $X$ with (continuous) distribution function $F$ (probably something should be known about the speed of convergence)?
I've looked for references on this topic, but I'm not sure what good search terms are for this question. All I have found are these papers, but they don't provide a satisfying answer:
A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables - J. Wellner
On the glivenko cantelli theorem for weighted empiricals based on independent random variables - R.S. Singh