An Abelian Group way to study Random Extended Intervals and their ARMA Processes

An extended interval is a range A = [A, A] where A may be bigger than A. This is not really natural but is what has been used as definition of extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product R×Z2, where Z2 = {0, 1} is the set of directions and the direction 0 is for increasing intervals and 1 for decreasing ones. For instance [3, 6]× {1} stands for the decreasing interval [6, 3]. Thereafter, we introduce on the set of extended intervals a family of metrics dγ, depending on a function γ(t), and show that there exists a unique metric dγ for which γ(t)dt is what we have called "adapted measure". This unique metric has very good properties, is simple to compute and has been implemented in the software R. Furthermore, we use this metric to define variability for random extended intervals. We further study extended interval-valued ARMA time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.