Almost sure invariance principle of $\beta-$mixing time series in Hilbert space

TL;DR

This paper establishes an almost sure invariance principle for stationary beta-mixing processes in Hilbert space, extending previous results to broader classes like Markov chains and autoregressive processes.

Contribution

It generalizes the invariance principle to Hilbert space-valued processes under beta-mixing, including Markov chains with Lyapunov conditions, using big-small blocks and embedding techniques.

Findings

Applicable to ergodic Markov chains

Valid for functional autoregressive processes

Extends previous invariance principles

Abstract

Inspired by \citet{Berkes14} and \citet{Wu07}, we prove an almost sure invariance principle for stationary $\beta-$mixing stochastic processes defined on Hilbert space. Our result can be applied to Markov chain satisfying Meyn-Tweedie type Lyapunov condition and thus generalises the contraction condition in \citet[Example 2.2]{Berkes14}. We prove our main theorem by the big and small blocks technique and an embedding result in \citet{gotze2011estimates}. Our result is further applied to the ergodic Markov chain and functional autoregressive processes.