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A maximal $ \mathbb{L}_{p}$-inequality for stationary sequences and its applications


Authors: Magda Peligrad, Sergey Utev and Wei Biao Wu
Journal: Proc. Amer. Math. Soc. 135 (2007), 541-550
MSC (2000): Primary 60F05, 60F17
DOI: https://doi.org/10.1090/S0002-9939-06-08488-7
Published electronically: August 8, 2006
MathSciNet review: 2255301
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Abstract: The paper aims to establish a new sharp Burkholder-type maximal inequality in $ \mathbb{L}_p$ for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.


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Additional Information

Magda Peligrad
Affiliation: Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025

Sergey Utev
Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, England
Email: sergey.utev@nottingham.ac.uk

Wei Biao Wu
Affiliation: Department of Statistics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: wbwu@galton.uchicago.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08488-7
Keywords: Martingale, maximal inequality, Markov chains, renewal sequences, Bernoulli shifts, invariance principle, stationary process
Received by editor(s): April 21, 2005
Received by editor(s) in revised form: August 31, 2005
Published electronically: August 8, 2006
Additional Notes: The first author was supported by an NSA grant.
The third author was supported by NSF grant DMS-0448704.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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