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$$X_t - \phi_1X_{t-1} - ... - \phi_pX_{t-p} = Z_t + \theta_{t-1} + ... + \theta_qZ_{t-q} $$  where $\{Z_t\}$ is a white noise process with zero mean and variance $\sigma^2$. For any autocovariance function $\gamma(h)$ for which $\lim_{h\to\infty}\gamma(h) = 0$ and any integer $k > 0$, there exists an ARMA(p,q) process with autocovariance function $\gamma_X(h)$ such that $\gamma_X(h) = \gamma(h)$ for all $h \leq k$. This makes the ARMA process a very useful tool in the analysis and modeling of many different time-series. (reword this blurb, too much directly from the text) Unfortunately, the phenomena we are observing are not discrete processes, so in order to get a proper understanding of the underlying physics, we need a continuous analog to the ARMA process. \!\!BE SURE TO CITE BROCKWELL AND DEVID TIME SERIES THEORY AND METHODS  **Quick %**Quick  intro to ARMA (an maybe a bit of stochastic analysis) ***Background, %***Background,  current use **White %**White  noise process -> "forcing term" -> ARMA **Stationary %**Stationary  process (requirement for ARMA and CARMA), define it ** %**  present equation ** %**  short hand with AR and MA polynomials (do this for CARMA too to make it easier to work with polynomials) **Autogressive %**Autogressive  behavior ***AR(1) %***AR(1)  process **Moving %**Moving  average behavior (show why stationary) **ARMA %**ARMA  is discrete, mention why we need a continuous method instead \subsection{CARMA}