The distribution of Rademacher sums

We find upper and lower bounds for Pr(Z ?x, > t), where x1, x2 . ... are real numbers. We express the answer in terms of the K-interpolation norm from the theory of interpolation of Banach spaces. INTRODUCTION Throughout this paper, we let e1 C 2' ... be independent Bernoulli random variables (that is, Pr(Ce = 1) = Pr(Ce = -1) = 2). We are going to look for upper and lower bounds for Pr(Z 8,7Xn > t), where xl, x2, ... is a sequence of real numbers such that x = (xn)n??= E 12 . Our first upper bound is well known (see, for example, Chapter II, ?59 of [5]): (1) Pr (Z nxn > t 1X112) ? However, if Ix II I Ixlll) =0. To look for lower bounds, we might first consider using some version of the central limit theorem. For example, using Theorem 7.1.4 of [2], it can be shown that for some constant c we have | ( ) 1 ,/2?? ~I 00 2/2 lX113 )3 Pr(E C n > t lIX11) f,27ds t ||X112) > CI ,( e-S210 ds c -2e _ /2 Cr 9 > x I2 -1 e-s2/2 d Received by the editors December 22, 1988 and, in revised form, August 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 60C05; Secondary 60G50.