ASYMPTOTIC-BEHAVIOR OF SELF-NORMALIZED TRIMMED SUMS - NONNORMAL LIMITS

• 成果类型: Article
• 署名作者: HAHN, MG; WEINER, DC
• 署名单位: Boston University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989937
• 发表日期: 1992
• 页码: 455-482
• 关键词: DISTRIBUTIONS

摘要

Let {X(j)} be independent, identically distributed random variables with continuous nondegenerate distribution F which is symmetric about the origin. Let {X(n)(1), X(n)(2), . . . , X(n)(n)} denote the arrangement of {X1, . . ., X(n)} in decreasing order of magnitude, so that with probability 1, \X(n)(1)\ > \X(n)(2)\ > ... > \X(n)(n)\. For integers r(n) --> infinity such that r(n)/n --> 0, define the self-normalized trimmed sum T(n) = SIGMA(i = r(n))nX(n)(i)/{SIGMA(i = r(n))nX(n)2(i)}1/2. The asymptotic behavior of T(n) is studied. Under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for T(n), various interesting nonnormal limit laws for T(n) are obtained and represented by means of infinite random series. In general, moreover, criteria for degenerate limits and stochastic compactness for {T(n)} are also obtained. Finally, more general results and technical difficulties are discussed.