Slow-explosive AR(1) processes converging to random walk

Abstract

This article investigates slow-explosive AR(1) processes, which converge to a random walk (RW) process with logarithm rates, to fill the gap between nearly non-stationary AR(1) and moderately deviated AR(1) processes, and derives the asymptotics of the least squares estimator using central limit theorems for (reduced) U-statistic. We successfully establish the smooth link between the nearly non-stationary AR(1) and the moderately deviated AR(1) processes. Some novel results are reported, which include the convergence of the least squares estimator to a biased fractional Brownian motion.

Keywords:

• Slow-explosive AR(1)
• U-statistic
• Random Walk

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62M10

Acknowledgement

We thank the referee for constructive suggestions which led to a substantial improvement in the revised version. SY Hwang’s work was supported by a grant from the National Research Foundation of Korea (NRF-2018R1A2B2004157). TY Kim’s work was supported by a grant from the National Research Foundation of Korea (NRF-2016R1D1A1B03934375).