arXiv 18 Oct 2026 · Mathematics — Statistics Theory
Existing regression inference methods for nonstationary heavy tailed data are largely limited to unit roots tests. Functional central limit theorems (FCLTs), which are commonly used for the asymptotic analysis of nonstationary models, are inapplicable to nonstationary data characterized by a memory parameter $d<1$ and a stability parameter $α<2$. Furthermore, it is well known that inference under nonstationarity is often non-conventional, and this issue is exacerbated in the presence of heavy-tailed data. The interaction of long memory and heavy tails significantly enlarges the nonstationary region---i.e., the covariate model space---where conventional inference methods typically fail. To address these challenges, we propose a new limit theory and methods that facilitate both workable asymptotics and conventional inference in regressions involving general nonstationary heavy-tailed covariate processes. Specifically, our approach develops a limit theory for heavy-tailed weakly nonstationary processes (hereafter, HT-WNPs), which lie on the threshold of nonstationarity (cf. Duffy and Kasparis, 2021, Annals of Statistics). We demonstrate that the proposed methods---including IVX-based and related instrumental-variable procedures---facilitate conventional inference for a broad class of stationary and nonstationary heavy-tailed covariates. This class includes nearly and mildly integrated arrays, as well as nonstationary fractional processes with memory parameter $1-1/α<d<2-1/α$, where $α\in(1,2]$. Our analysis is applicable to both parametric and kernel regression models. The finite-sample performance of the proposed methods is evaluated through a comprehensive simulation study. Additionally, we provide an empirical application to the predictability of stock returns using long-memory heavy-tailed predictors.