POINT PROCESS AND PARTIAL SUM CONVERGENCE FOR WEAKLY DEPENDENT RANDOM-VARIABLES WITH INFINITE VARIANCE

• 成果类型: Article
• 署名作者: DAVIS, RA; HSING, TL
• 署名单位: Texas A&M University System; Texas A&M University College Station
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988294
• 发表日期: 1995
• 页码: 879-917
• 关键词: random vectors

摘要

Let {xi(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 = xi(1) + ... + xi(n), suitably normalized, when {xi(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 {xi(j)/a(n), j = 1,..., n}, where a(n) is the 1 - n(-1) quantile of the distribution of \xi(1)\. Then for 0 < alpha < 1 (- alpha is the exponent of regular variation), S-n is asymptotically stable if N-n converges weakly, and for 1 less than or equal to < 2, the same is true under a condition that is slightly stronger than the weak convergence of N-n. We also consider large deviation results for S-n. In particular, we show that for any sequence of constants {t(n)} satisfying nP[xi(1) > t(n)] --> 0, P[S-n > t(n)]/(nP[xi(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.