On probabilities of large deviations in some classes of k-dimensional Borel sets

Let X1, X2,..., be a sequence of independent identically distributed random vectors in k-dimensional Euclidean space Rk and let [Phi](A) be the standard normal distribution in Rk, Sn = X1 + ... + Xn. In this paper a behavior of a relation P{1/[radical sign]nSn [set membership, variant] A}/[Phi](A) when set A is contained in some class of Borel sets and [Phi](A) --> 0, n --> [infinity], is investigated. Particularly, the conditions are obtained which are necessary and sufficient for 40 uniformly in all sets A which are the differences between convex Borel sets in Rk satisfying the condition 62 Here [Lambda](z) is a function such that [Lambda](z) [short up arrow] [infinity], [Lambda](z)/z[var epsilon]0 [downwards arrow] 0, 0 < [var epsilon]0 < 1.