Donsker's theorem for self-normalized partial sums processes

• Miklós Csörgő ^[1] ; Barbara Szyszkowicz ^[1] ; Qiying Wu ^[2]
  1. [1] Carleton University

     Carleton University

     Canadá

     
  2. [2] Australian National University

     Australian National University

     Australia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 3, 2003, págs. 1228-1240
• Idioma: inglés
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• Resumen

  • Let X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if max1≤j≤n|Xj|/Vn→P0, as n→∞, where V2n=∑nj=1X2j.