• Process Convergence of Self-Normalized Sums of i.i.d. Random Variables Coming from Domain of Attraction of Stable Distributions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/123/01/0085-0100

• # Keywords

Domain of attraction; process convergence; self-normalized sums; stable distributions.

• # Abstract

In this paper we show that the continuous version of the self-normalized process $Y_{n,p}(t)=S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p},0 &lt; t \leq 1;p&gt;0$ where $S_n(t)=\sum^{[nt]}_{i=1}X_i$ and $V_{(n,p)}=\left(\sum^n_{i=1}|X_i|^p\right)^{1/p}$ and $X_i i.i.d.$ random variables belong to $DA(\alpha)$, has a non-trivial distribution $\mathrm{iff } p=\alpha=2$. The case for $2&gt;p&gt;\alpha$ and $p\leq\alpha &lt; 2$ 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 Csörgő et al. who showed Donsker’s theorem for $Y_{n,2}(\cdot p)$, i.e., for $p=2$, holds $\mathrm{iff } \alpha=2$ and identified the limiting process as a standard Brownian motion in sup norm.

• # Author Affiliations

1. Stat-Math Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India
2. Department of Statistics, Presidency University, 87/1 College Street, Kolkata 700 073, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019

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