A data-dependent DKW inequality for regenerative Markov chains
We prove a version of the Dvoretzky-Kiefer-Wolfowitz inequality for Markov chains with a regenerative structure. Suppose we have a regenerative Markov chain with stationary distribution $\pi$. Given a functional $\theta$ on the state space and a confidence level $1-\delta$, our result provides a uniform $1-\delta$ confidence band for the CDF of $\theta$ under $\pi$ based on the empirical CDF. By inversion, we get a $1-\delta$ confidence band for the quantile function of $\theta$ under $\pi$. Our bounds are fully explicit and nearly optimal. In addition, they are data-dependent in the following sense: in the formula for the width of the confidence band, the leading term can be computed directly from the sample path without any a priori information about the convergence rate of the chain. A convergence bound is required, but it contributes to the width of the confidence band only through a lower-order term. For this reason, our result is attractive for Markov chains whose convergence rate is much quicker in practice than what can be proved in theory. Data-dependent bounds of this type are called empirical concentration inequalities in the literature. Thus, our result is an empirical concentration inequality for the empirical CDF of $\theta$ given the sample path.