Abstract: We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted L1-distance. Using equivalent forms of this metric and the specific choice of excursion levels, we formulate the prediction problem as a minimization of a certain target functional which involves the excursion metric. Existence of the solution and weak consistency of the predictor are discussed. An application to the extrapolation of stationary heavy-tailed random functions illustrates the use of the aforementioned theory. Numerical experiments with the prediction of Gaussian, α- and max-stable random functions show the practical merits of the approach.
Joint work with A. Das and V. Makogin
 A. Das, V. Makogin, and E. Spodarev. Extrapolation of stationary random fields via level sets. Theory of Probability and Mathematical Statistics, 106:85–103, 2022.
 V. Makogin and E. Spodarev. Prediction of random variables by excursion metric projections. Preprint, arXiv:2207.00447v2, September 2022. https://doi.org/10.48550/arXiv.2207.00447.