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

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arxiv 2209.12535 v2 pith:HA3EWW7Q submitted 2022-09-26 math.PR

Jianya Lu , Wei Biao Wu , Zhijie Xiao , Lihu Xu This is my paper · ORCID
classification math.PR
keywords citetresultalmostappliedberkes14beta-chaincondition
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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.

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Cited by 2 Pith papers

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    For dependent, approximately stationary random functions in C0, the partial sum process is within O(N^{-τ}) of a functional Brownian motion in Prokhorov and Wasserstein distance.

  2. Choosing the Right Norm for Change Point Detection in Functional Data

    math.ST 2025-01 conditional novelty 7.0 of 10

    An L1-norm based change point test for functional time series is introduced, with bootstrap validity, theoretical power comparisons, and a power enhancement against sparse alternatives.

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