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We introduce a new class of autoregressive models for spherical time series,where the dimension of the spheres on which the observations of the time seriesare situated may be finite-dimensional or infinite-dimensional as in the caseof a general Hilbert sphere. Spherical time series arise in various settings.We focus here on distributional and compositional time series. Applying asquare root transformation to the densities of the observations of adistributional time series maps the distributional observations to the Hilbertsphere, equipped with the Fisher-Rao metric. Likewise, applying a square roottransformation to the components of the observations of a compositional timeseries maps the compositional observations to a finite-dimensional sphere,equipped with the geodesic metric on spheres. The challenge in modeling suchtime series lies in the intrinsic non-linearity of spheres and Hilbert spheres,where conventional arithmetic operations such as addition or scalarmultiplication are no longer available. To address this difficulty, we considerrotation operators to map observations on the sphere. Specifically, weintroduce a class of skew-symmetric operator such that the associatedexponential operators are rotation operators that for each given pair of pointson the sphere map one of the points to the other one. We exploit the fact thatthe space of skew-symmetric operators is Hilbertian to develop autoregressivemodeling of geometric differences that correspond to rotations of spherical anddistributional time series. Motivating data for our methods include a timeseries of yearly observations of bivariate distributions of the minimum/maximumtemperatures for a period of 120 days during each summer for the years1990-2018 at Los Angeles (LAX) and John F. Kennedy (JFK) internationalairports.

Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series