On weighted approximations in $D[0,1]$ with applications to self-normalized partial sum processes

Abstract

Let $X,X_1,X_2,...$ be a sequence of non-degenerate i.i.d. random variables with 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 $S_n={\sum }_{j=1}^n{X}_j$, under the assumption that $X$ belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes $\{S_{[nt]}/V_n,0\le t\le 1\}$, where $V_n^2={\sum }_{j=1}^n{X}_j^2$. $L_p$ approximations of self-normalized partial sum processes are also discussed.

22 pages