The approximation of partial sums of $\phi$-mixing random variables with values in a Banach space $B$ by a $B$-valued Brownian motion is obtained. This result yields the compact as well as the functional law of the iterated logarithm for these sums. As an application we strengthen a uniform law of the iterated logarithm for classes of functions recently obtained by Kaufman and Philipp (1978). As byproducts we obtain necessary and sufficient conditions for an almost sure invariance principle for independent identically distributed $B$-valued random variables and an almost sure invariance principle for sums of $d$-dimensional random vectors satisfying a strong mixing condition.
J. Kuelbs. Walter Philipp. "Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables." Ann. Probab. 8 (6) 1003 - 1036, December, 1980. https://doi.org/10.1214/aop/1176994565