Semigroup stationary processes and spectral representation

Abstract

We present an extended definition of the second-order stationarity concept. This is based on the theory of harmonic analysis for semigroups with involution. It provides a spectral representation for a wide class of processes which are non-stationary in the usual weak sense, and allows miscellaneous spectral representation results to be unified. Many applications are given to illustrate the concept. Most of these are already known, %as symmetric, locally %stationary, stationary reducible by space transformation, %multiplicative-stationary processes or processes with independent %increments. but some are new, such as the multiplicative-symmetric processes. We are less concerned with proving fundamental results than with opening up a new field of investigation for spectral representation of non-stationary processes.