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

TL;DR

This paper investigates optimal weighted approximations of partial sum processes and self-normalized processes in the space of cadlag functions, providing new results under normal attraction conditions and discussing $L_p$ approximations.

Contribution

It introduces new weighted approximation results for partial sum and self-normalized processes in $D[0,1]$, extending understanding under the domain of attraction of the normal law.

Findings

Established best possible weighted approximations for partial sum processes.

Extended results to self-normalized partial sum processes.

Discussed $L_p$ approximation methods for these 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 $S_n=\sum_{j=1}^nX_j$, 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 $V_n^2=\sum_{j=1}^nX_j^2$. $L_p$ approximations of self-normalized partial sum processes are also discussed.