PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 109, Number 2, June 1990 THE DISTRIBUTION OF RADEMACHER SUMS S. J. MONTGOMERY-SMITH (Communicated by William D. Sudderth) Abstract. We find upper and lower bounds for Pr(23 +-Xn > t), where x, , x2, ... are real numbers. We express the answer in terms of the ^-interpolation norm from the theory of interpolation of Banach spaces. Introduction Throughout this paper, we let e{, e2, ... be independent Bernoulli random variables (that is, Pr(e/J = 1) = Pr(en = -1) = \). We are going to look for upper and lower bounds for Pr(£] enxn > t), where xx, x2, ... is a sequence of real numbers such that x = (xn)^=x e l2. Our first upper bound is well known (see, for example, Chapter II, §59 of [5]): d) pr(Ee^>'W2)<^'2/2- However, if ||x||, < oo, this cannot also provide a good lower bound, because then we have another upper bound: (2) Pr(EV->IMIi)-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 Thus, for some constant c we have that if r < c~ (log||x||3/||x||2) , then /•oo 2 ~2 _'2/2 Pr(£e^>'Nl2)>^7, e~s/2ds>C-^-t-. Received by the editors December 22, 1988 and, in revised form, August 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 60C05; Secondary 60G50. Key words and phrases. Rademacher sum, Holmstedt's formula. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 517 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use