When is the student t-statistic asymptotically standard normal?

• 成果类型: Article
• 署名作者: Gine, E; Gotze, F; Mason, DM
• 署名单位: University of Connecticut; University of Connecticut; University of Bielefeld; University of Delaware
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1514-1531
• 关键词: sums

摘要

Let X, X-i, i is an element of N, be independent, identically distributed random variables. It is shown that the Student t-statistic based upon the sample {X-i}(i less than or equal to 1)(n) is asymptotically N(0, 1) if and only if X is in the domain of attraction of the normal law. It is also shown that, for any X, if the self-normalized sums U-n := Sigma(i=1)(n) X-i/(Sigma(i=1)(n) X-i(2))(1/2), n is an element of N, are stochastically bounded then they are uniformly subgaussian that is, sup(n) E exp(lambda U-n(2)) < infinity for some lambda > 0.