X(t) — a X(t — 1) —a2X(t— 2) = (1), (12.9) 
we can introduce the vector process X(t) with two elements X), X2 as 
X(t) = X(t) 
(12.10) 
X1(t) = a2X(t— 1)(= a2X2(t- 1). 
Then Eq. 12.9 can be written 
X2(t) 1 a, | | X2¢-D 1 
X() X(t-1) 
Here setting 
Xy(t) 0 a2 0 
= X(t), =a, €(t) = €(f), 
X0(t) (t) aie By \e (t) = €(¢) 
Eq. 12.11 becomes 
X(t) = aX(t-1)+€(0). (12.12) 
This transformation shows that AR(2) can be inverted into a Markovian linear process or 
a vector process X(t) = [Xj(t), X2(t)]’. 
By the same procedure, the autoregressive process of order n, AR(n), or more 
generally the autoregressive moving average model ARMA (an, m), can be shown to be 
invertible into a Markovian process, introducing the matrix of inverting functions. 
The characteristics of a Markovian process contribute to the derivation of distribu- 
tion functions, applying the Fokker—Planck equation, and also for the application of the 
Kalman filters. 
12.4 THE FOKKER-PLANCK EQUATION 
To show the behavior of the transitional probability density function p, of a stochas- 
tic process as a Markov process, formally a linear second order partial differential 
equation called the Fokker—Planck equation is used. This equation was developed by 
Fokker (1914) and Planck (1917), to indicate the Brownian motion of molecules, and it 
has also been utilized as the equation of motion to show the behavior of the transitional 
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