The following inner-outer type factorization is obtained for the sequence space ℓp: if the complex sequence F = (F0, F1,F2,...) decays geometrically, then for an p sufficiently close to 2 there exists J and G in ℓp 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 ℓp; and G is a cyclic vector for S on ℓp.
These ideas are used to show that and ARMA equation with characteristic roots inside and outside of the unit circle has Symmetric-α-Stable solution,s 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 autregressive representation of the process is similarly obtained.
Copyright © 2016 Elsevier Inc.
The definitive version is available at: https://www.sciencedirect.com/science/article/pii/S0022247X16000329
Cheng, Raymond, and William T. Ross. "An Inner–outer Factorization in ℓ P with Applications to ARMA Processes." Journal of Mathematical Analysis and Applications 437, no. 1 (2016): 396-418. doi:10.1016/j.jmaa.2016.01.009.
Cheng, Raymond and Ross, William T., "An inner-outer factorization in ℓp with applications to ARMA processes" (2016). Math and Computer Science Faculty Publications. 187.