A general stability theorem for $B$-valued random variables is obtained which refines a result of Kuelbs and Zinn. Our proof is based on two exponential inequalities for sums of independent $B$-valued r.v.'s essentially due to Yurinskii and appears particularly simple. We then use our theorem to prove strong invariance principles, LIL results and other related stability results for sums of i.i.d. $B$-valued r.v.'s in the domain of attraction of a Gaussian law. Most of these results seem to be still unknown for real-valued r.v.'s.
"Stability Results and Strong Invariance Principles for Partial Sums of Banach Space Valued Random Variables." Ann. Probab. 17 (1) 333 - 352, January, 1989. https://doi.org/10.1214/aop/1176991512