  By well known results of probability theory, any sequence of random variables
with bounded second moments has a subsequence satisfying the central limit
theorem and the law of the iterated logarithm in a randomized form. In this
paper we give criteria for a sequence $(X_n)$ of random variables to have a
subsequence $(X_{n_k})$ whose weighted partial sums, suitably normalized,
converge weakly to a stable distribution with parameter $0<\alpha<2$.
