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

Authors: Victor H. de la Pena, Michael J. Klass, Tze Leung Lai

Published: 2004 (Journal Paper)

Source: Annals of Probability

Algorithm: Self-Normalized Processes

arXiv: math/0410102

DOI: 10.1214/009117904000000397

Summary

Derives exponential inequalities and moment bounds for self-normalized random variables and martingales, with a focus on laws of the iterated logarithm. The central device is an exponential supermartingale Y_t(λ) built from a process A_t and its self-normalizer B_t; when Y_t(λ) ≤ 1 in expectation or is a supermartingale, tight tail and moment bounds follow. The paper also establishes a bounded LIL for general adapted sequences centered at truncated conditional expectations.

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 A_t, let Y_t(λ)=exp{λ A_t-λ^2 B_t^2/2}. We develop inequalities for the moments of A_t/B_t or sup_{t≥0} A_t/{B_t(\log\log B_t)^{1/2}} and variants thereof, when EY_t(λ)≤1 or when Y_t(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 A_t=M_t and B_t=\sqrt{<M>_t}, and sums of conditionally symmetric variables d_i with A_t=\sum_{i=1}^t d_i 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 R^m, 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 \sum_{i=1}^t d_i and \sum_{i=1}^t d_i^2.

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arXiv Abstract math/0410102 arXiv PDF math/0410102 arXiv HTML math/0410102 DOI 10.1214/009117904000000397

Tags

  • Self-normalized processes

  • Martingales

  • Law of the iterated logarithm

  • Exponential inequalities

  • Probability theory

  • Supermartingales

  • Sequential analysis