An inner-outer factorization in ℓp with applications to ARMA processes
The following inner-outer type factorization is obtained for the sequence space ℓ p: if the complex sequence F=(F0, F1, F2,...) decays geometrically, then for and p sufficiently close to 2 there exist J and G in ℓ p such that F = J * G; J is orthogonal in the Birkhoff-James sense to all of its forward shifts SJ, S2J, S3J,...; J and F generate the same S-invariant subspace of ℓ p; and G is a cyclic vector for S on ℓ p.
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 autoreressive representation of the process is similarly obtained.