Exponential inequalities for sums of random vectors

This paper presents some generalizations of S. N. Bernstein's exponential bounds on probabilities of large deviations to the vector case. Inequalities for probabilities of large deviations of sums of independent random vectors are derived under a Cramér's type restriction on the rate of growth of absolute moments of the summands. Estimates are obtained for random vectors with values in Banach space, Sharper bounds hold in the case of finite-dimensional Euclidean or separable Hilbert spaces.