Prediction of random variables by excursion metric projections

We use the concept of excursions for the prediction of random variables without any moment existence assumption. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted $L^1$-distance. Using equivalent forms of this metric and a specific choice of excursion levels, we formulate the prediction problem as a minimization of a certain target functional. Existence and uniqueness of the solution 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 $\alpha$-stable time series and random fields round up the paper.