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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.2. LINEAR PROCESS 8<br />

Finally, I provide examples using geophysical series.<br />

2.2 Linear Process<br />

It should be stressed that any wide sense stationary random process can be rep-<br />

resented <strong>via</strong> the superposition <strong>of</strong> a deterministic part sn plus a non deterministic<br />

part xn (random or stochastic part),<br />

zn = sn + xn. (2.1)<br />

Wold decomposition theorem (Wold, 1965; Ulrych and Sacchi, 2005), in addition<br />

states that the non deterministic part can be represented as a filtered sequence <strong>of</strong><br />

white noise<br />

∞<br />

xn = giεn−i<br />

i=1<br />

(2.2)<br />

where g0 = 1, ∞ i=1 |gi| 2 < ∞ (finite length impulse response) and εn represents the<br />

white noise which is uncorrelated with sn. I can change the equation above by a<br />

finite length discrete general linear model<br />

N<br />

xn = giεn−i<br />

i=1<br />

(2.3)<br />

where εn is the innovations process and xn depends on its past input values (there-<br />

fore it is casual).

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