Self-normalized large deviations

• 成果类型: Article
• 署名作者: Shao, QM
• 署名单位: University of Oregon
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 285-328
• 关键词: erdos-renyi; LAWS; sums

摘要

Let {X, X-n, n greater than or equal to 1} be a sequence of independent and identically distributed random variables. The classical Cramer-Chernoff large deviation states that lim(n-->infinity) n(-1) ln P((Sigma(i=1)(n) X-i)/n greater than or equal to x) = ln rho(x) if and only if the moment generating function of X is finite in a right neighborhood of zero. This paper uses n((p-1)/p)V(n,p) = n((p-1)/p)(Sigma(i=1)(n) \X-i\(p))(1/p) (p > 1) as the normalizing constant to establish a self-normalized large deviation without any moment conditions. A self-normalized moderate deviation, that is, the asymptotic probability of P(S-n/V-n,V-p greater than or equal to x(n)) for x(n) = o(n((p-1)/p)), is also found for any X in the domain of attraction of a normal or stable law. As a consequence, a precise constant in the self-normalized law of the iterated logarithm of Griffin and Kuelbs is obtained. Applications to the limit distribution of self-normalized sums, the asymptotic probability of the t-statistic as well as to the Erdos-Renyi-Shepp law of large numbers are also discussed.