each shock affects X values infinitely far into the future from when they occur, any given value Xt is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression (where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as When the polynomial division on the right side is carried out, the polynomial in the backshift operator applied to has an infinite order—that is, an infinite number of lagged values of appear on the right side of the equation. Characteristic polynomial The autocorrelation function of an AR(p) process can be expressed as where are the roots of the polynomial where B is the backshift operator, where is the function defining the autoregression, and where are the coefficients in the autoregression. The formula is valid only if all the roots have multiplicity 1. The autocorrelation function of an AR(p) process is a sum of decaying exponentials. Each real root contributes a component to the autocorrelation function that decays exponentially. Similarly, each pair of complex conjugate roots contributes an exponentially damped oscillation. Graphs of AR(p) processes The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise. For an AR(1) process with a positive , only the previous term in the process and the noise term contribute to the output. If is close to 0, then the process still looks like white noise, but as approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a low pass filter. For an AR(2) process, the previous two terms and the noise term contribute to the output. If both and are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If is positive while is negative, then the process favors changes in sign between terms of the process. The output oscillates. This can be likened to edge detection or detection of change in direction. Example: An AR(1) process An AR(1) process is given by:where is a white noise process with zero mean and constant variance . (Note: The subscript on has been dropped.) The process is weak-sense