  Series of univariate distributions indexed by equally spaced time points are
ubiquitous in applications and their analysis constitutes one of the challenges
of the emerging field of distributional data analysis. To quantify such
distributional time series, we propose a class of intrinsic autoregressive
models that operate in the space of optimal transport maps. The autoregressive
transport models that we introduce here are based on regressing optimal
transport maps on each other, where predictors can be transport maps from an
overall barycenter to a current distribution or transport maps between past
consecutive distributions of the distributional time series. Autoregressive
transport models and their associated distributional regression models specify
the link between predictor and response transport maps by moving along
geodesics in Wasserstein space. These models emerge as natural extensions of
the classical autoregressive models in Euclidean space. Unique stationary
solutions of autoregressive transport models are shown to exist under a
geometric moment contraction condition of Wu and Shao (2004), using properties
of iterated random functions. We also discuss an extension to a varying
coefficient model for first order autoregressive transport models. In addition
to simulations, the proposed models are illustrated with distributional time
series of house prices across U.S. counties and annual summer temperature
distributions.
