# Adjustment needed for multivariate Dvoretzky–Kiefer–Wolfowitz inequality on MCMC samples?

I was thinking about studying bounds on the multivariate empirical cumulative distribution function for samples from an MCMC chains. The multivariate Dvoretzky–Kiefer–Wolfowitz inequality would seem almost fit for this purpose except it assumes IID samples whereas my samples will be Markovian.

I think that any difference due to assuming IID vs Markov property will vanish. The number of adjacent random variables in the chain is $$n-1$$ while the number of pairs between random variables is $$\frac{n^2-n}{2}$$. Their ratio in terms of the sample size $$n \geq 2$$ gives

$$\frac{n-1}{\frac{n^2-n}{2}} = \frac{2}{n}$$

which indicates that even if adjacent samples are statistically dependent we will still have $$\lim_{n \rightarrow \infty} \frac{2}{n} = 0.$$

Most of the chains I run are $$n \geq 2000$$, so this fraction for me is $$\frac{2}{n} \leq \frac{1}{1000}$$ which does subjectively seem pretty small to me.

But even so, assuming the Markov property instead of IID, is there an adjusted preasymptotic bound I can use for CDF over a collection of sampled instances of my parameters? Or am I falsely assuming there should exist an adjustment?