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.
Categories: math.PR