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

Abstract

For $(X_t)$ a two-sided $\alpha$-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 $\boldsymbol{X}_t=(X_{t-m},\ldots,X_t,X_{t+1},\ldots,X_{t+h})$, $m\ge0$, $h\ge1$, 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 $\{\boldsymbol{s}\in\mathbb{R}^{m+h+1}: \hspace{0.3cm} \|\boldsymbol{s}\|=1\}$ for $\|\cdot\|$ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors $\boldsymbol{X}_t$ 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 $\boldsymbol{X}_t$ 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.