Sharp oracle inequalities and universality of the AIC and FPE

TL;DR

This paper proves that AIC and FPE criteria are universally efficient for a wide class of dependent processes, extending their optimality beyond linear models with independent innovations.

Contribution

The paper establishes sharp oracle inequalities for AIC and FPE under weak dependence, demonstrating their universality and asymptotic efficiency across diverse dynamical systems.

Findings

AIC and FPE are optimal under general weak dependence conditions.

The results apply to various complex systems including Garch models and dynamical systems.

AIC and FPE exhibit universal properties beyond linear processes.

Abstract

In two landmark papers, Akaike introduced the AIC and FPE, demonstrating their significant usefulness for prediction. In subsequent seminal works, Shibata developed a notion of asymptotic efficiency and showed that both AIC and FPE are optimal, setting the stage for decades-long developments and research in this area and beyond. Conceptually, the theory of efficiency is universal in the sense that it (formally) only relies on second-order properties of the underlying process $(X_t)_{t\in \mathbb{Z}}$, but, so far, almost all (efficiency) results require the much stronger assumption of a linear process with independent innovations. In this work, we establish sharp oracle inequalities subject only to a very general notion of weak dependence, establishing a universal property of the AIC and FPE. A direct corollary of our inequalities is asymptotic efficiency of these criteria. Our framework contains many prominent dynamical systems such as random walks on the regular group, functionals of iterated random systems, functionals of (augmented) Garch models of any order, functionals of (Banach space valued) linear processes, possibly infinite memory Markov chains, dynamical systems arising from SDEs, and many more.