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Self-Normalized Processes: Exponential Inequalities, Moment Bounds and Iterated Logarithm Laws

Victor H. de la Peña, Michael J. Klass and Tze Leung Lai
The Annals of Probability
Vol. 32, No. 3, A (Jul., 2004), pp. 1902-1933
Stable URL: http://www.jstor.org/stable/3481599
Page Count: 32
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Self-Normalized Processes: Exponential Inequalities, Moment Bounds and Iterated Logarithm Laws
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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 $B_{t}>0$ and At, let Yt(λ)= exp{λ At-λ 2Bt 2/2}. We develop inequalities for the moments of At/Bt or supt≥ 0At/{Bt( log log Bt)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=1 tdi 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=1 tdi and ∑i=1 tdi 2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

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