Title: Probability, Statistics and Planet Earth, I: Geotemporal covariances,
Abstract: The study of covariances (or positive definite functions) on the sphere (the
Earth, in our motivation) goes back to Bochner and Schoenberg (1940--42) and to
the first author (1969, 1973), among others. Extending to the geotemporal case
(sphere cross line, for position and time) was for a long time an obstacle to
geostatistical modelling. The characterisation question here was raised by the
authors and Mijatović in 2016, and answered by Berg and Porcu in 2017.
Extensions to multiple products (of spheres and lines) follows similarly
(Guella, Menegatto and Peron, 2016). We survey results of this type, and
related applications e.g. in numerical weather prediction.