Limit theorems for sums of random exponentials

• 成果类型: Article
• 署名作者: Ben Arous, G; Bogachev, LV; Molchanov, SA
• 署名单位: New York University; University of Leeds; University of North Carolina; University of North Carolina Charlotte
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0406-3
• 发表日期: 2005
• 页码: 579-612
• 关键词: energy

摘要

We study limiting distributions of exponential sums S-N( t) = Sigma(N)(i=1) e(tXi) as t --> infinity, N --> infinity, where (X-i) are i. i. d. random variables. Two cases are considered: (A) ess sup X-i = 0 and ( B) ess sup X-i = infinity. We assume that the function h(x) = - log P{X-i > x} (case B) or h( x) = - logP{X-i > rho < infinity} ( case A) is regularly varying at 8 with index 1 < rho < infinity(case B) or 0 < rho < infinity(case A). The appropriate growth scale of N relative to t is of the form e(lambda H0(t)) (0 < lambda < infinity), where the rate function H-0(t) is a certain asymptotic version of the function H( t) = log E[e(tXi)] (case B) or H( t) = - logE[e(tXi)] (case A). We have found two critical points, lambda(1) < lambda(2), below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < lambda