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

Sebastien Fries

Research output: Working paper / PreprintWorking paperProfessional


For (Xt) 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 Xt=(Xt−m,…,Xt,Xt+1,…,Xt+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∈Rm+h+1:∥s∥=1} for ∥⋅∥ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors Xt 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 (Xt) will have to be "anticipative enough" for Xt 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.
Original languageEnglish
Number of pages75
Publication statusPublished - Sept 2018


  • Noncausal process
  • Anticipative process
  • Multivariate stable distribution
  • Pattern recognition
  • Prediction
  • Forecasting
  • Spectral representation


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