An inner–outer factorization in ℓp with applications to ARMA processes

The following inner–outer type factorization is obtained for the sequence space ℓpℓp: if the complex sequence F=(F0,F1,F2,…)F=(F0,F1,F2,…) decays geometrically, then for any p sufficiently close to 2 there exist J and G in ℓpℓp such that F=J⁎GF=J⁎G; J is orthogonal in the Birkhoff–James sense to all of its forward shifts SJ,S2J,S3J,…SJ,S2J,S3J,… ; J and F generate the same S  -invariant subspace of ℓpℓp; and G is a cyclic vector for S   on ℓpℓp. These ideas are used to show that an ARMA equation with characteristic roots inside and outside of the unit circle has Symmetric-α-Stable solutions, in which the process and the given white noise are expressed as causal moving averages of a related i.i.d. SαS white noise. An autoregressive representation of the process is similarly obtained.