Probabilities of large deviations for sums of i.i.d. Banach space valued random variables are investigated when the laws of the random variables converge weakly and a uniform exponential integrability condition is satisfied. Furthermore, a discussion of possible improvements of the estimates is given, when the probability is estimated that the sum lies in a convex set.
"On the Probability of Large Deviations in Banach Spaces." Ann. Probab. 12 (2) 427 - 435, May, 1984. https://doi.org/10.1214/aop/1176993298