  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
$L^1$-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, $\alpha$-stable and further heavy--tailed time series round up the
paper.
