A functional central limit theorem for regenerative chains

TL;DR

This paper proves a functional central limit theorem for certain stochastic chains with memory decay, providing explicit variance formulas and applying results to binary autoregressive and Ising chains.

Contribution

It establishes a new FCLT for chains with summable memory decay and identifies the limiting variance under stronger decay conditions.

Findings

FCLT holds for chains with summable memory decay

Explicit variance formulas are derived for specific processes

Applications include binary autoregressive and power-law Ising chains

Abstract

Using the regenerative scheme of Comets, Fern\'andez and Ferrari (2002), we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the FCLT is fulfilled.