The famed DKW inequality states the following:
$$\mathbb{P}\left(\sup_{x\in\mathbb{R}}|F_n(x)-F(x)|>\epsilon\right)\leq 2e^{-2n\epsilon^2}$$.
Further using Bahadur's representation, we naturally can get Woodruff type confidence intervals for all corresponding quantiles without needing to estimate the density function. In such case, at quantile $p$, the asymptotic variance for the sample quantile is $\frac{p(1-p)}{nf(\xi_p)}$, here $\xi_p$ denotes the $p$-th quantile value for $F(x)$.
However, it only holds for i.i.d. samples.
Suppose we have a cluster sampling scenario: clusters are sampled i.i.d. at first, and then all observations with in each cluster are all taken as samples. Specifically, we have $K$ clusters, for cluster $i=1,\cdots,K$, there are a total of $N_i$ observations each with value $Y_{ij}$, $j=1,\cdots,N_i$. Supposably we can treat observations that are not in the same cluster independent, but not with in each cluster. $N_i$ are not all the same and can be assumed to follow another distribution.
My question is, can we establish a similar DKW type inequality in this case?