Anomalous Fluctuations in Autoregressive Models with Long-Term Memory

TL;DR

This paper investigates an autoregressive model with long-term memory, revealing how its fluctuations behave differently over small and large timescales and how the memory kernel influences this behavior.

Contribution

It introduces a detailed numerical analysis of an autoregressive process with a power-law memory kernel, highlighting its self-affine fractal-like behavior and the impact of the memory exponent.

Findings

Root-mean-square displacement follows a power law at small times

Displacement saturates at large times

Exponent varies with the memory kernel's power exponent

Abstract

An autoregressive model with a power-law type memory kernel is studied as a stochastic process that exhibits a self-affine-fractal-like behavior for a small time scale. We find numerically that the root-mean-square displacement for the time interval increases with a power law for small time but saturates at sufficiently large time. The exponent changes with the power exponent of the memory kernel.