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

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.