Without explicit reference to these conditions the statio- 
narity of ARMA processes is declared if the stationarity 
condition is satisfied: Every ARMA process is stationary if 
all zeros of A(z), called the poles of the ARMA process, 
are placed inside the unit circle of the complex plane. 
Non-stationarity is often caused by a trend function in the 
time series which may be determined and removed. A trend 
can be removed, for instance, by using derivatives of the 
time series /4/. A time series is called an autoregressive 
integrated moving average (ARIMA) process of order (p,d.q), 
if the d-th derivatives of the series are stationary and 
can be described by an ARMA (p,q) process. 
3. Further Representations of ARMA Processes 
The theory of ARMA processes is related to other concepts 
which are used for describing and analyzing time series. 
Especially the linear filtering technique, the spectral 
analysis and the methods using autocorrelation functions 
may be considered as further representations of ARMA 
processes. These relations are derived in the following. 
3.1 Linear Filtering 
Linear filtering techniques describe the output: x(t) of a 
linear filter by  convoluting the input e(t) with the 
impulse response of the filter ht). 
x(t) = h(t) % e(t) = Z h(i)-e(t-1i) 
Transforming this equation from the time/space domain into 
the complex z-domain applying the z-transform, the 
convolution operation becomes a simple multiplication 
X(z).-.H(z) -.E(z) 
In terms of linear filter technique an ARMA process may be 
considered as the output of a linear feedback-feedfront 
system with a white noise process as input /14/. The trans- 
formed impulse response, called the transfer function, is 
described by the polynomials in z of the ARMA process 
H(z) = D(z) / Az) 
3.2 Spectral Analysis 
Many applications of time series analysis are based on 
spectral analysis. A function x(t) defined in the time 
domain can be transformed in a function X(u) defined in the 
frequency domain by applying the Fourier transformation 
XK(u):;- FU x(tc) ) 
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