  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.
