Random Additive Perturbation of a \(k\) Term Recurrence Relation

Abstract

This paper deals with a real valued process driven by a general \(k\) term recurrence relation and perturbed by an additive random term. The first step is the case when the perturbation is a fixed constant \(\theta \). To get a transformation - which depends on this constant term - of a \(k\) dimensional space and apply classical techniques [24], we imbed the recurrence relation in \({\mathbb{R}}^{k}\) in a particular way, implying the use of a small parameter. We then study the transfer operator of the transformation, taking care of the uniformity with respect to the constant \(\theta \). The second step is the study of the perturbed process. Following [1], we introduce the expectation of the individual \(k\)-dimensional transfer operators studied before. This “averaged” transfer operator has a finite number of modulus 1 eigenvalues and a spectral gap. It admits an absolutely continuous invariant measure (ACIM) on \({\mathbb{R}}^{k}\), which, in turn, gives an ACIM for the initial real valued process.