Predicting Random Walks and a Data-Splitting Prediction Region

Perhaps the first nonparametric, asymptotically optimal prediction intervals are provided for univariate random walks, with applications to renewal processes. Perhaps the first nonparametric prediction regions are introduced for vector-valued random walks. This paper further derives nonparametric data-splitting prediction regions, which are underpinned by very simple theory. Some of the prediction regions can be used when the data distribution does not have first moments, and some can be used for high-dimensional data, where the number of predictors is larger than the sample size. The prediction regions can make use of many estimators of multivariate location and dispersion.