Prediction of random variables by excursion metric projections

TL;DR

This paper introduces an excursion metric-based approach for predicting random variables without relying on moment assumptions, demonstrating its effectiveness through theoretical analysis and numerical experiments on various heavy-tailed processes.

Contribution

It develops a novel excursion metric framework for prediction, enabling analysis without moment conditions, and applies it to heavy-tailed and Gaussian time series.

Findings

Existence of predictors under the excursion metric

Weak consistency of the proposed predictor

Successful application to heavy-tailed time series

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 $L^1$-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.