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

Raymond Cheng
William T. Ross, University of Richmond

Abstract

The following inner-outer type factorization is obtained for the sequence space ℓ ^p: if the complex sequence F=(F₀, F_1, 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.