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Moderate Deviations for Random Sums of Heavy-Tailed Random Variables
作者姓名:Fu  Qing  GAO
作者单位:School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China
基金项目:Research supported by NsFC (No. 10271091, 10571139) The author thanks Professor Cline for his preprint Cline and Hsing [9] and thanks the anonymous referee for his suggestions and comments.
摘    要:Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E(X1) and let {N(t); t ≥0} be a random process taking non-negative integer values with finite mean λ(t) = E(N(t)) and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P((X1 +… +XN(t)) -λ(t)μ 〉 x) uniformly for x ∈γb(t), ∞) are obtained, where γ〉 0 and b(t) can be taken to be a positive function with limt→∞ b(t)/λ(t) = 0.

关 键 词:大偏差  泊松过程  随机变量  分布函数
收稿时间:29 December 2003
修稿时间:2003-12-29

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables
Fu Qing GAO.Moderate Deviations for Random Sums of Heavy-Tailed Random Variables[J].Acta Mathematica Sinica,2007,23(8):1527-1536.
Authors:Fu Qing Gao
Affiliation:(1) School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China
Abstract:Let {X n ; n ≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E(X 1) and let {N(t); t ≥ 0} be a random process taking non-negative integer values with finite mean λ(t) = E(N(t)) and independent of {X n ; n ≥ 1}. In this paper, asymptotic expressions of P((X 1+⋯+X N(t))−λ(t)μ > x) uniformly for xγb(t),∞) are obtained, where γ > 0 and b(t) can be taken to be a positive function with lim t →∞ b(t)/λ(t) = 0. Research supported by NSFC (No. 10271091, 10571139)
Keywords:large deviations  moderate deviations  extended regular variation  Poisson process
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