Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variableswith mean zero. The best possible weighted approximations are investigated in$D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$, where$S_n=\sum_{j=1}^nX_j$, under the assumption that $X$ belongs to the domain ofattraction of the normal law. The conclusions then are used to establishsimilar results for the sequence of self-normalized partial sum processes$\{S_{[nt]}/V_n, 0\le t\le 1\}$, where $V_n^2=\sum_{j=1}^nX_j^2$. $L_p$approximations of self-normalized partial sum processes are also discussed.
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机译:令$ X,X_1,X_2,... $是一个非退化i.i.d序列。均值为零的随机变量。对于部分和过程$ \ {S _ {[nt]},0 \ le t \ le 1 \} $,在$ D [0,1] $中研究了最佳可能的加权逼近,其中$ S_n = \ sum_ {j = 1} ^ nX_j $,假设$ X $属于正常法则的吸引力域。然后,这些结论可用于为自标准化的部分和过程$ \ {S _ {[nt]} / V_n,0 \ le t \ le 1 \} $的序列建立相似的结果,其中$ V_n ^ 2 = \ sum_ { j = 1} ^ nX_j ^ 2 $。还讨论了自归一化部分和过程的$ L_p $近似值。
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