Examples of time series with long-range dependence can be found in many sciences. In this talk, we explore statistical inference for distribution functions under long-memory. Our interest lies in nonparametric methods, and in particular kernel smoothing for estimation, and the empirical moment generating function for testing. One particular problem is inference for distribution functions that change over time. This has implications for changes in quantiles, which are of high relevance, for example for issues like climate change. We discuss asymptotic results and give some numerical examples using real and simulated data.