Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws.

• Victor H. de la Peñ ^[1] ; Michael J. Klass ^[3] ; Tze Leung Lai ^[2]
  1. [1] Columbia University

     Columbia University

     Estados Unidos

     
  2. [2] Stanford University

     Stanford University

     Estados Unidos

     
  3. [3] University of California at Berkeley

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 3, 1, 2004, págs. 1902-1933
• Idioma: inglés
• DOI: 10.1214/009117904000000397
• Texto completo no disponible (Saber más ...)
• Resumen

  • Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt>0 and At, let Yt(λ)=exp{λAt−λ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt≥0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)≤1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At=Mt and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables di with At=∑i=1tdi and $B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}$ . A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in Rm, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑i=1tdi and ∑i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.