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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 14

ARMA Processes

Autoregressive-moving-average (ARMA) processes are linear processes that only depend on a finite number of parameters, which facilitates their use in statistics (see section 10.3).

14.1. Autoregressive processes

DEFINITION 14.1.– images is said to be a autoregressive process of order p (AR(p)) if:

[14.1] images

with πp ≠ 0, and where (εt) is a white noise such that:

images

Uniqueness of the decomposition.– If there exists a weakly stationary process (Xt) satisfying [14.1], then the decomposition is unique. Indeed, if:

images

then we have:

images1

Then, if image, we have images and, by stationarity, images, images. Hence, ...

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