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

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dc.creatorde la Pena, Victor H.
dc.creatorKlass, Michael J.
dc.creatorLai, Tze Leung
dc.date2004-10-05
dc.date.accessioned2026-07-07T05:12:53Z
dc.date.available2026-07-07T05:12:53Z
dc.descriptionSelf-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{λA_t-λ^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq 0}A_t/{B_t(\log \log B_{t})^{1/2}} and variants thereof, when EY_t(λ)\leq 1 or when Y_t(λ) is a supermartingale, for all λbelonging 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}^td_i and B_t=\sqrt\sum_{i=1}^td_i^2. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in R^m, m\ge 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}^td_i and \sum_{i=1}^td_i^2.
dc.descriptionPublished by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000397
dc.identifierhttps://arxiv.org/abs/math/0410102
dc.identifierhttp://arxiv.org/abs/math/0410102
dc.identifierAnnals of Probability 2004, Vol. 32, No. 3A, 1902-1933
dc.identifierdoi:10.1214/009117904000000397
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/72746
dc.subjectProbability
dc.subject60E15, 60G42, 60G44 (Primary) 60G40. (Secondary)
dc.titleSelf-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws
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