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

Authors
  • BASAK, GOPAL K1
  • BISWAS, ARUNANGSHU2
  • 1 Indian Statistical Institute, Stat-Math Unit, Kolkata, 700 108, India , Kolkata (India)
  • 2 Presidency College, Deptartment of Statistics, Kolkata, 700 073, India , Kolkata (India)
Type
Published Article
Journal
Proceedings - Mathematical Sciences
Publisher
Springer India
Publication Date
Feb 20, 2013
Volume
123
Issue
1
Pages
85–100
Identifiers
DOI: 10.1007/s12044-013-0109-8
Source
Springer Nature
Keywords
License
Yellow

Abstract

In this paper we show that the continuous version of the self-normalized process Yn,p(t) = Sn(t)/Vn,p + (nt − [nt])X[nt] + 1/Vn,p,0 < t ≤ 1; p > 0 where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_n(t)=\sum_{i=1}^{[nt]} X_i$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$\end{document} and Xi i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p = α = 2. The case for 2 > p > α and p ≤ α < 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 Yn,2(·), i.e., for p = 2, holds iff α = 2 and identified the limiting process as a standard Brownian motion in sup norm.

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