Suppose Xt obeys Equation (35d), but with the stipulation that the Dl terms are real-valued constants, while the φl terms are independent RVs, each having a uniform distribution on the interval (−π, π]. Show that the process so defined is in fact a harmonic process and that its ACVS is given by
Three realizations from this harmonic process for the case of µ = 0, L = 1, f1 = 1/20 and D1 = √ 2 are shown in the right-hand column of Figure 36. This choice of parameters yields an ACVS that is identical to the Gaussian-distributed harmonic process behind the realizations in the left-hand column. Note that each realization has a fixed amplitude so that Xt is restricted to the interval [− √ 2, √ 2] both from one realization to the next and within a given realization. The fact that Xt is bounded implies that it cannot have a Gaussian distribution.
Our second example can be called a “random phase” harmonic process and suggests that random phases are more fundamental than random amplitudes in formulating a stationary process (see Exercise [2.19] for another connection between stationarity and a process whose realizations can also be thought of as “randomly phased” versions of a basic pattern). In fact, in place of our more general formulation via Equation (35c), some authors define a harmonic process as having fixed amplitudes and random phases (see, e.g., Priestley, 1981). In practical applications, however, a random phase harmonic process is often sufficient – particularly since we would need to have multiple realizations of this process in order to distinguish it from one having both random amplitudes and phases. (Exercise [2.20] explores the consequence of allowing the random phases to come from a nonuniform distribution.) Note that any given realization from a harmonic process can be regarded as a deterministic function since the realizations of the RVs in right-hand sides of either Equation (35c) or (35d) determine Xt for all t. Assuming the frequencies fl and mean µ are known, knowledge of any segment of a realization of Xt with a length at least as long as 2L is enough to fully specify the entire realization. In one sense, randomizing the phases (or amplitudes and phases) is a mathematical trick allowing us to treat models like Equation (35d) within the context of the theory of stationary processes. In another sense, however, all stationary processes can be written as a generalization of a harmonic process with an infinite number of terms – as we shall see, this is the essence of the spectral representation theorem to be discussed in Chapter 4 (harmonic analysis itself is studied in more detail in Chapter 10, and simulation of harmonic processes is discussed in Section 11.1).
It should be noted that the independence of the φl RVs is a sufficient condition to ensure stationarity (as demonstrated by an assumption of just un correlated ness does not suffice in general). Although independence of phases is often assumed for convenience when the random phase model described by is fit to actual data, this assumption can lead to subtle problems. Walden and Prescott (1983) modeled tidal elevations using the random phase harmonic process approach. The disagreement (often small) between the calculated and measured tidal PDFs is largely attributable to the lack of phase independence of the constituents of the tide (see Section 10.1 for details).