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\end {cases} $$  are known as {\em white noise processes} and are written as $Z_{t} \sim WN(0,\sigma^{2})$. We can use a white noise as a forcing term to build a useful set of linear difference equations to analyze a more complicated time-series.  There exist a class of finite difference equations used in the analysis of discrete time series known as autoregressive-moving average (ARMA) processes. These processes give allow  us a way of inspecting the behavior to quantify properties  of a  time-series with a simple but thoroughly descriptive parametric structure. A stationary process $\{X_t\}$ can be modeled by an ARMA(p,q) process if at every time $t$ $$X_t - \phi_1X_{t-1} - ... - \phi_pX_{t-p} = Z_t + \theta_{t-1} + ... + \theta_qZ_{t-q} $$ 
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  $$d^{P}f(t) + \alpha_{P-1} d^{P-1}f(t) + ... + \alpha_{0} f(t) = \beta_{Q}d^{Q}w(t) + \beta_{Q-1} d^{Q-1}w(t) + ... + w(t)$$    Where $d$ represents a change in a variable between times $t$ and $t + dt$, $f$ represents the state of the system minus the mean, and $w \sim WN(0, \sigma^{2})$ continuous-time white noise random process representing the driving noise in the system due to non-linear effects. In this case, it will represent temperature fluctuations due to magnetohydrodynamic instabilities in the accretion disk. In order for the process to be stationary, stationary we must require that  $p < q$. The most well known example is the case where $p=1$ and $q=0$ CAR(1) process also known in the astronomical community as a damped random walk (DRW).   We construct the auto-regressive polynomial as the characteristic polynomial of the auto-regressive side of the CARMA process.