Donsker's theorem for self-normalized partial sums processes
Donsker's theorem for self-normalized partial sums processes
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DOI：
10.1214/aop/1055425777
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发表时间：
2003-07
影响因子：
2.3
通讯作者：
Qi Wu;B. Szyszkowicz;M. Csörgo
中科院分区：
文献类型：
--
作者：
Qi Wu;B. Szyszkowicz;M. Csörgo
设X，X1，X2，.为非退化同分布列.均值为零的随机变量在本文中，我们证明了一个自规范化版本的Donsker定理成立的假设下，X属于正常的法律吸引域。由此产生的弧正弦定律的延伸也进行了讨论。我们还建立了一个弱不变性原理对独立随机变量的自正规化、自随机化部分和过程成立，当且仅当max 1 ≤j≤n，且假设它们是关于均值零对称的|XJ|/Vn→P0，当n→∞时，其中V2 n =∑nj= 1X 2 j。
Let X,X1,X2,… 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 max1≤j≤n|Xj|/Vn→P0, as n→∞, where V2n=∑nj=1X2j.