The angular momentum evolution of stars close to massive black holes (MBHs) is driven by secular torques. In contrast to two-body relaxation, where interactions between stars are incoherent, the resulting resonant relaxation (RR) process is characterized by coherence times of hundreds of orbital periods. In this paper, we show that all the statistical properties of RR can be reproduced in an autoregressive moving average (ARMA) model. We use the ARMA model, calibrated with extensive N-body simulations, to analyze the long-term evolution of stellar systems around MBHs with Monte Carlo simulations. We show that for a single-mass system in steady state, a depression is carved out near an MBH as a result of tidal disruptions. Using Galactic center parameters, the extent of the depression is about 0.1pc, of similar order to but less than the size of the observed hole in the distribution of bright late-type stars. We also find that the velocity vectors of stars around an MBH are locally not isotropic. In a second application, we evolve the highly eccentric orbits that result from the tidal disruption of binary stars, which are considered to be plausible precursors of the S-stars in the Galactic center. We find that RR predicts more highly eccentric (e > 0.9) S-star orbits than have been observed to date.