PUB - Publikationen an der Universität Bielefeld

Let X be a random variable with probability distribution P-X concentrated on [-1,1] and let Q(x) be a polynomial of degree k greater than or equal to 2. The characteristic function of a random variable Y = Q(X) is of order O(1/\t\(1/k)) as \t\ --> infinity if P-X is sufficiently smooth. In addition, for every epsilon: 1/k > epsilon > 0 there exists a singular distribution P-X such that every convolution P-X(n star) is also singular while the characteristic function of Y is of order O(1/\t\(1/k-epsilon)). While the characteristic function of X is small when "averaged," the characteristic function of the polynomial transformation Y of X is uniformly small.