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July 2004 Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws
Victor H. de la Peña, Michael J. Klass, Tze Leung Lai
Ann. Probab. 32(3): 1902-1933 (July 2004). DOI: 10.1214/009117904000000397


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{λAtλ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt0At/{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, m1, 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.


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Victor H. de la Peña. Michael J. Klass. Tze Leung Lai. "Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws." Ann. Probab. 32 (3) 1902 - 1933, July 2004.


Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1075.60014
MathSciNet: MR2073181
Digital Object Identifier: 10.1214/009117904000000397

Primary: 60E15 , 60G42 , 60G44
Secondary: 60G40

Keywords: Inequalities‎ , iterated logarithm , Martingales , Self-normalized

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • July 2004
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