Prediction of random variables by excursion metric projections
Vitalii Makogin, Evgeny Spodarev · Bernoulli · 2025
We use the concept of excursions to predict random variables without moment existence assumptions. To do so, an excursion metric on the space of random variables is defined as 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. Existence of the solution and weak consistency of the predictor are discussed. An application to extrapolating stationary heavy-tailed random functions illustrates the use of our approach. Numerical experiments predicting Gaussian, α-stable and further heavy–tailed time series complement the paper.