Darling--Erdős theorem for self-normalized sums

• Miklós Csörgő ^[1] ; Barbara Szyszkowicz ^[1] ; Qiying Wang ^[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. 2, 2003, págs. 676-692
• Idioma: inglés
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• Resumen

  • Let X,X1,X2,… be i.i.d. nondegenerate random variables, Sn=∑nj=1Xj and V2n=∑nj=1X2j. We investigate the asymptotic \vspace*{1pt} behavior in distribution of the maximum of self-normalized sums, max1≤k≤nSk/Vk, and the law of the iterated logarithm for self-normalized sums, Sn/Vn, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling--Erdős-type theorem as well as an Erdős--Feller--Kolmogorov--Petrovski-type test for self-normalized sums.