Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

2.2. LINEAR PROCESS 8
Finally, I provide examples using geophysical series.
2.2 Linear Process
It should be stressed that any wide sense stationary random process can be rep-
resented via the superposition of a deterministic part sn plus a non deterministic
part xn (random or stochastic part),
zn = sn + xn. (2.1)
Wold decomposition theorem (Wold, 1965; Ulrych and Sacchi, 2005), in addition
states that the non deterministic part can be represented as a filtered sequence of
white noise
∞
xn = giεn−i
i=1
(2.2)
where g0 = 1, ∞ i=1 |gi| 2 < ∞ (finite length impulse response) and εn represents the
white noise which is uncorrelated with sn. I can change the equation above by a
finite length discrete general linear model
N
xn = giεn−i
i=1
(2.3)
where εn is the innovations process and xn depends on its past input values (there-
fore it is casual).