Consistency of Akaike's Information Criterion for Infinite Variance Autoregressive Processes

Abstract

Suppose $\{X_n\}$ is a $p$th order autoregressive process with innovations in the domain of attraction of a stable law and the true order $p$ unknown. The estimate $\hat{p}$ of $p$ is chosen to minimize Akaike's information criterion over the integers $0, 1, \cdots, K$. It is shown that $\hat{p}$ is weakly consistent and the consistency is retained if $K \rightarrow \infty$ as $N \rightarrow \infty$ at a certain rate depending on the index of the stable law.