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In this paper, we focus on isotropic and stationary sphere-cross-time randomfields. We first introduce the class of spherical functionalautoregressive-moving average processes (SPHARMA), which extend in a naturalway the spherical functional autoregressions (SPHAR) recently studied in [8,7]; more importantly, we then show that SPHAR and SPHARMA processes ofsufficiently large order can be exploited to approximate every isotropic andstationary sphere-cross-time random field, thus generalizing to thisinfinite-dimensional framework some classical results on real-valued stationaryprocesses. Further characterizations in terms of functional spectralrepresentation theorems and Wold-like decompositions are also established.

SPHARMA approximations for stationary functional time series on the sphere