Auto-regressive process

A stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139801.png" /> whose values satisfy an auto-regression equation with certain constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139802.png" />:

(*)

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139804.png" /> is some positive number and where the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139805.png" /> are usually assumed to be uncorrelated and identically distributed around their average value 0 with a variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139806.png" />. If all the zeros of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139807.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139808.png" /> lie inside the unit circle, then equation (*) has the solution

For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398013.png" /> be a white noise process with spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398014.png" />; in such a case the only kind of auto-regressive process satisfying equation (*) will be a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398015.png" /> with spectral density

which is stationary in the wide sense if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398017.png" /> has no real zeros. The autocovariances (cf. Autocovariance) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398018.png" /> of the process satisfy the recurrence relation

and, in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398020.png" />, have the form

The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398022.png" /> of the auto-regression are connected with the auto-correlation coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398023.png" /> of the process by the matrix relation

[1]	U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957)
[2]	E.J. Hannan, "Time series analysis" , Methuen , London (1960)

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From The Encyclopedia of Math: Auto-regressive process.

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