Process convergence of self normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

TL;DR

This paper investigates the convergence behavior of self-normalized sums of i.i.d. random variables from the domain of attraction of stable distributions, establishing conditions under which non-trivial limits exist.

Contribution

It extends previous results by characterizing the convergence of self-normalized processes for different p-values and stable domain parameters, identifying when non-degenerate limits occur.

Findings

Non-trivial distribution only when p=α=2.

Elimination of convergence for 2 > p > α and p ≤ α < 2.

Extension of Donsker's theorem to broader settings.

Abstract

In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(\alpha)$, has a non trivial distribution iff $p=\alpha=2$. The case for $2 > p > \alpha$ and $p \le \alpha < 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Cs\"org\"o et al. who showed Donsker's theorem for $Y_{n,2}(\cdot)$, i.e., for $p=2$, holds iff $\alpha =2$ and identified the limiting process as standard Brownian motion in sup norm.