  The main result of this paper is a general central limit theorem for
distributions defined by certain renewal type equations. We apply this to
weakly self-avoiding random walks. We give good error estimates and Gaussian
tail estimates which have not been obtained by other methods. We use the lace
expansion and at the same time develop a new perspective on this method: We
work with a fixed point argument directly in the x-space without using Laplace
or Fourier transformation.
