A NONSTANDARD LAW OF THE ITERATED LOGARITHM FOR TRIMMED SUMS
成果类型：
Article
署名作者：
HAEUSLER, E
署名单位：
University of Munich
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176989270
发表日期：
1993
页码：
831-860
关键词：
摘要：
Let X(i), i greater-than-or-equal-to 1, be independent random variables with a common distribution in the domain of attraction of a strictly stable law, and for each n greater-than-or-equal-to 1 let X1, n less-than-or-equal-to ... less-than-or-equal-to X(n, n) denote the order statistics of X1, ..., X(n). In 1986, S. Csorgo, Horvath and Mason showed that for each sequence k(n), n greater-than-or-equal-to 1, of nonnegative integers with k(n) --> infinity and k(n)/n --> 0 as n --> infinity, the trimmed sums S(n)(k(n)) = X(kn + 1,n) + ... + X(n - k(n - kn, n) converge in distribution to the standard normal distribution, when properly centered and normalized, despite the fact that the entire sums X1 + ... +X(n) have a strictly stable limit, when properly centered and normalized. The asymptotic almost sure behavior of S(n)(k(n)) strongly depends on the rate at which k(n) converges to infinity The sequences k(n) approximately c log log n as n --> for 0 < c < infinity constitute a borderline case between a classical law of the iterated logarithm and a radically different behavior. This borderline case is investigated in detail for nonnegative summands X(i).
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