Almost sure invariance principle of <i>β</i> -mixing time series in Hilbert space

Inspired by Berkes et al. [3] and Wu [37], we prove an almost sure invariance principle for stationary (Formula presented.) -mixing stochastic processes defined on Hilbert spaces. Our result applies to Markov chains satisfying the Meyn-Tweedie-type Lyapunov condition, thereby generalizing the contraction condition in Berkes et al. [3, Example 2.2]. The main theorem is established using the big and small blocks technique and an embedding result from Götze and Zaitsev [14]. Examples of ergodic Markov chains and functional autoregressive processes are provided as applications.