Path prediction of aggregated alpha-stable moving averages using semi-norm representations

For (X_t) a two-sided α-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form X_t=(X_t−m,…,X_t,X_t+1,…,X_t+h), m≥0, h≥1, are multivariate alpha-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders -sets {s∈R^m+h+1:∥s∥=1} for ∥⋅∥ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors X_t when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and (X_t) will have to be "anticipative enough" for X_t to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. The approach extends to processes resulting from the linear combination of stable moving averages and applied to several examples.