A tight Gaussian bound for weighted sums of Rademacher random variables

摘要

Let epsilon(1) . . . . . epsilon(n) be independent identically distributed Rademacher random variables, that is P{epsilon(i) = +/- 1) = 1/2. Let S-n =a(1)epsilon(1) + . . . +a(n)epsilon(n). where a = (a(1) . . . . . a(n)) is an element of R" is a vector such that a(1)(2) + . . . + a(n)(2) <= 1. We find the smallest possible constant c in the inequality