We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.
Herold Dehling. Walter Philipp. "Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables." Ann. Probab. 10 (3) 689 - 701, August, 1982. https://doi.org/10.1214/aop/1176993777