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

#### Abstract

The following inner-outer type factorization is obtained for the sequence space ℓ^{ p}: if the complex sequence **F**=(*F _{0}, F_{1,} F_{2},...) *decays geometrically, then for and

*p*sufficiently close to 2 there exist

**J**and

**G**in ℓ

*such that*

^{p}**F**=

**J * G**;

**J**is orthogonal in the Birkhoff-James sense to all of its forward shifts

*S*

**J**

*,*

**S**^{2}**J**

*,*

**S**^{3}**J**,

*...;*

**J**and

**F**generate the same

*S-*invariant subspace of ℓ

*; and*

^{p}**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.

*This paper has been withdrawn.*