Convergence to a Stable Distribution Via Order Statistics

Let $X_1, X_2, \cdots$ be i.i.d. random variables whose common distribution function $F$ is in the domain of attraction of a nonnormal stable distribution. A simple, probabilistic proof of the convergence of the normalized partial sums to the stable distribution is given. The proof makes use of an elementary property of order statistics and clarifies the manner in which the largest few summands determine the limiting distribution. The method is applied to determine the limiting distribution of self-norming sums and deduce a representation for the limiting distribution. The representation affords an explanation of the infinite discontinuities of the limiting densities which occur in some cases. Application of the technique to prove weak convergence in a separable Hilbert space is explored.