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Let {Xj} be an i.i.d. sequence of Banach space valued r.v.'s and let Sn = Z, 1 Xj . For certain positive sequences bn oo, we determine the exact asymptotic behavior of E exp{(b2/n)D(Sn/bn)}, where D is a smooth function. We also prove a large deviation principle for {2Y(Sn/bn)} .

Moderate deviations and associated Laplace approximations for sums of independent random vectors