In this section, the strict-sense, second-order wide-sense, and higher-order characterizations of GACS stochastic processes are presented in the time domain. Moreover, the second-order spectral characterization is discussed. Examples of GACS processes and motivations to adopt such a model are provided. See (Izzo and Napolitano 1998a) and (Izzo and Napolitano 2002a) for a treatment in the nonstochastic or functional approach.
Definition 2.2.1 The real-valued process , is said to be with almost-periodic structure in the strict sense if its Nth-order distribution function
is almost-periodic in t (in one of the senses considered in Section 1.2), for every fixed .
Thus, for a process with almost-periodic structure in the strict sense, the Nth-order distribution function can be expressed by the (generalized) Fourier series
where is a countable set depending on and and