Weak convergence of self-normalized partial sums processes

Abstract

Let $\{X,X_n,n\ge 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0,S_n={\sum }_{i=1}^n{X}_i$ and $V_n^2={\sum }_{i=1}^n{X}_i^2,n\ge 1.$ A weak convergence theorem is established for the self-normalized partial sums processes $\{S_{[nt]}/V_n,0\le t\le 1\}$ when $X$ belongs to the domain of attraction of a stable law with index $\alpha \in (0,2]$. The respective limiting distributions of the random variables ${\max }_{1\le i\le n}|X_i|/S_n$ and ${\max }_{1\le i\le n}|X_i|/V_n$ are also obtained under the same condition.