Donsker's theorem for self-normalized partial sums processes
成果类型：
Article
署名作者：
Csörgö, M; Szyszkowicz, B; Wang, QY
署名单位：
Carleton University; Australian National University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
2003
页码：
1228-1240
关键词：
摘要：
Let X, X-1, X-2,... be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if max(1less than or equal tojless than or equal ton) \X-j\/V-n --> (P) 0, as n --> infinity, where V-n(2) = Sigma(j=1)(n) X-j(2).