Lacunary series and stable distributions

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 $0<\alpha <2$.