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There exist a class of finite difference equations used in the analysis of discrete time series known as autoregressive-moving average (ARMA) processes. Given any autocovariance function $\gamma(h)$ for which $\lim_{h\to\infty}\gamma(h) = 0$ and for 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 These processes give us  a very useful tool in way of inspecting  the analysis and modeling behavior  of time-series. time-series with a simple parametric structure.  A stationary process $\{X_t\}$ is an ARMA(p,q) process if at every time $t$ $$X_t - \phi_1X_{t-1} - ... - \phi_pX_{t-p} = Z_t + \theta_{t-1} + ... + \theta_qZ_{t-q} $$  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)  \!\!BE SURE TO CITE BROCKWELL AND DEVID TIME SERIES THEORY AND METHODS  **Quick intro to ARMA (an maybe a bit of stochastic analysis)  ***Background, current use