Tail Adversarial Stability for Regularly Varying Linear Processes and their Extensions

TL;DR

This paper investigates tail adversarial stability in regularly varying linear processes, extending the concept to stochastic volatility, max-linear models, and transformations, with implications for limit theorems in statistics.

Contribution

It verifies tail adversarial stability for additive linear processes and explores its extensions and invariance properties, filling a gap in the existing literature.

Findings

Tail adversarial stability holds for regularly varying additive linear processes.

Extensions to stochastic volatility and max-linear models are established.

Invariance under monotone transformations is demonstrated.

Abstract

The notion of tail adversarial stability has been proven useful in obtaining limit theorems for tail dependent time series. Its implication and advantage over the classical strong mixing framework has been examined for max-linear processes, but not yet studied for additive linear processes. In this article, we fill this gap by verifying the tail adversarial stability condition for regularly varying additive linear processes. We in addition consider extensions of the result to a stochastic volatility generalization and to a max-linear counterpart. We also address the invariance of tail adversarial stability under monotone transforms. Some implications for limit theorems in statistical context are also discussed.