MODERATE DEVIATIONS AND ASSOCIATED LAPLACE APPROXIMATIONS FOR SUMS OF INDEPENDENT RANDOM VECTORS

Let {X(j)} be an i.i.d. sequence of Banach space valued r.v.'s and let S(n) = SIGMA(j = 1)n X(j). For certain positive sequences b(n) --> infinity, we determine the exact asymptotic behavior of E exp{(b(n)2/n)PHI(S(n)/b(n)}, where PHI is a smooth function. We also prove a large deviation principle for {L(S(n)/b(n))}.