Mathematical Statistics and Stochastic Processes

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

[14.1]

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

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

then we have:

1

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