Probabilities of moderate deviations in a Banach space

Abstract

(1) P[ S,1Vn/ > tn ] 1 (D(tn) = of -tn) P[ Snl / 0. In this case, called moderate deviations by them, they proved that (1) holds without existence of a moment generating function; it suffices that E?XiK2+c+S 0. Also in this case they proved that EIXjIP < oo, Vp < 2 + c, is a necessary condition for (1). Recently Slastnikov (1978) has shown that, again with tn = Ac log n , (1) holds if and only if X1 satisfies the tail-condition