$\mathrm{E}_{\gamma}$-Mixing Time

We investigate the mixing times of Markov kernels under $\mathsf{E}_{\gamma}-\mathbf{divergence}$. We demonstrate that the zero-error $\mathsf{E}_{\gamma}- \mathbf{mixing}$ time, for any $\gamma > 1$, of irreducible and aperiodic Markov chains, is bounded, a property that is not shared by the TV-mixing time. We further obtain upper bounds on the $\mathsf{E}_{\gamma}-\mathbf{mixing}$ times for a broad family of contractive Markov kernels via a new non-linear strong data processing inequality for the $\mathsf{E}_{\gamma}- \mathbf{divergence}$. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data.