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70 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSNow let b = ωt and c = λ and use (2.101) to represent (2.100) asq t = a cos(ωt + λ)= cos(ωt)[a cos(λ)] − sin(ωt)[a sin(λ)].Next, build the time-series q t = q 1t + q 2t from the two sub-series q 1t andq 2t ,whereforj =1, 2q jt =cos(ω j t)[a j cos(λ j )] − sin(ω j t)[a j sin(λ j )],and ω 1 < ω 2 . The result is a periodic function which is displayed ontheleftsideofFigure2.3.10.80.60.40.20-0.2-0.4-0.6-0.8-11 6 11 16 21 26 31 363020100-10-20-301 6 11 16 21 26 31 36Figure 2.3: For 0 ≤ ω 1 < ··· < ω N ≤ π, q t = P Nj=1 q jt ,whereq jt =cos(ω j t)[a j cos(λ j )] − sin(ω j t)[a j sin(λ j )]. Left panel: N =2. Rightpanel: N =1000The composite process with N = 2 is clearly deterministic but ifyou build up the analogous series with N = 100 of these components,as shown in the right panel of Figure 2.3, the series begins to look likea random process. It turns out that any stationary random process canbe arbitrarily well approximated in this fashion letting N →∞.