Weak convergence of self-normalized partial sums processes

TL;DR

This paper proves a weak convergence theorem for self-normalized partial sums processes of i.i.d. variables in the domain of attraction of a stable law, and derives limiting distributions for certain maxima ratios.

Contribution

It establishes new weak convergence results for self-normalized sums and maxima ratios when variables are in the domain of attraction of a stable law.

Findings

Weak convergence of self-normalized partial sums processes.

Limiting distributions for maxima ratios of variables.

Results hold for stable law domain of attraction with index (0,2].

Abstract

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ 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.