Autoregressive Processes on Riemannian Manifolds

This paper introduces a Riemannian autoregressive (R-AR) model of order one, generalising classical discrete-time stochastic processes to manifold-valued data. The model is based on two parameters, a parameter $\mu$ representing the intrinsic central tendency as the Fr\'echet mean and an autoregressive parameter $\phi$ controlling the stationarity and ergodic properties. Due to the inherent dependence structure of the R-AR process, the estimation procedure for these parameters necessitates new asymptotic results for dependent processes on manifolds. Thus, we establish a strong law of large numbers for the sample Fr\'echet mean set of ergodic Markov chains in proper metric spaces. By proving this general consistency result, we move beyond the limitations of classical i.i.d. theory to provide the mathematical foundation required for the strong consistency of our proposed estimators. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.