The Annals of Probability

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

Victor H. de la Peña, Michael J. Klass, and Tze Leung Lai

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Abstract

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.

Article information

Source
Ann. Probab., Volume 32, Number 3 (2004), 1902-1933.

Dates
First available in Project Euclid: 14 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1089808415

Digital Object Identifier
doi:10.1214/009117904000000397

Mathematical Reviews number (MathSciNet)
MR2073181

Zentralblatt MATH identifier
1075.60014

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Martingales self-normalized inequalities iterated logarithm

Citation

de la Peña, Victor H.; Klass, Michael J.; Leung Lai, Tze. Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab. 32 (2004), no. 3, 1902--1933. doi:10.1214/009117904000000397. https://projecteuclid.org/euclid.aop/1089808415


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