  We propose a new auto-regressive model for the statistical analysis of
multivariate distributional time series. The data of interest consist of a
collection of multiple series of probability measures supported over a bounded
interval of the real line, and that are indexed by distinct time instants. The
probability measures are modelled as random objects in the Wasserstein space.
We establish the auto-regressive model in the tangent space at the Lebesgue
measure by first centering all the raw measures so that their Fr\'echet means
turn to be the Lebesgue measure. Using the theory of iterated random function
systems, results on the existence, uniqueness and stationarity of the solution
of such a model are provided. We also propose a consistent estimator for the
model coefficient. In addition to the analysis of simulated data, the proposed
model is illustrated with two real data sets made of observations from age
distribution in different countries and bike sharing network in Paris. Finally,
due to the positive and boundedness constraints that we impose on the model
coefficients, the proposed estimator that is learned under these constraints,
naturally has a sparse structure. The sparsity allows furthermore the
application of the proposed model in learning a graph of temporal dependency
from the multivariate distributional time series.
