Theorems of large deviations in the multivariate invariance principle

Let B be a real separable Banach space with norm ßB, X, X1, X2, ... be a sequence of centered independent identically distributed random variables taking values in B. Let sn = sn(t), 0 <= t <= 1 be the random broken line such that sn(0) = 0, sn(k/n) = n-1/2 [Sigma]i=1k Xi for n = 1, 2, ... and k = 1, ..., n. Denote snB = sup0 <= t <= 1 sn(t)B and assume that w(t), 0 <= t <= 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(snB > r) = P(wB > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension.