Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance

Let {ξ j } be a strictly stationary sequence of random variables with regularly varying tail probabilities. We consider, via point process methods, weak convergence of the partial sums, S n = ξ 1 +... + ξ n , suitably normalized, when {ξ j } satisfies a mild mixing condition. We first give a characterization of the limit point processes for the sequence of point processes N n with mass at the points {ξ j /a n , j = 1 n}, where a n is the 1 - n -1 quantile of the distribution of |ξ 1 |. Then for 0 t n ] →0, P[S n > t n ]/(nP[ξ 1 > t n ]) tends to a constant which can in general be different from 1. Applications of our main results to self-norming sums, m-dependent sequences and linear processes are also given.