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Laws of the iterated logarithm for self-normalised Lévy processes at zero


Authors: Boris Buchmann, Ross A. Maller and David M. Mason
Journal: Trans. Amer. Math. Soc. 367 (2015), 1737-1770
MSC (2010): Primary 60G51, 60F10, 60F15, 60G44
DOI: https://doi.org/10.1090/S0002-9947-2014-06112-6
Published electronically: October 9, 2014
MathSciNet review: 3286497
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Abstract: We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as $ t\downarrow 0$) for the ``self-normalised'' process $ (X_{t}-at)/\sqrt {V_{t}}$, $ t>0$, constructed from a Lévy process $ (X_{t})_{t\geq 0}$ having quadratic variation process $ (V_{t})_{t\geq 0}$, and an appropriate choice of the constant $ a$. We apply them to obtain LILs when $ X_{t}$ is in the domain of attraction of the normal distribution as $ t\downarrow 0$, when $ X_{t}$ is symmetric and in the Feller class at 0, and when $ X_{t}$ is a strictly $ \alpha -$stable process. When $ X_{t}$ is attracted to the normal distribution, an important ingredient in the proof is a Cramér-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.


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Additional Information

Boris Buchmann
Affiliation: Institute of Mathematical Sciences & School of Finance and Applied Statistics, Australian National University, Canberra ACT 0200, Australia
Email: Boris.Buchmann@anu.edu.au

Ross A. Maller
Affiliation: Centre for Mathematical Analysis and its Applications, Australian National University, Canberra ACT 0200, Australia
Email: Ross.Maller@anu.edu.au

David M. Mason
Affiliation: Department of Applied Economics and Statistics, University of Delaware, 206 Townsend Hall, Newark, Delaware 19717
Email: davidm@Udel.Edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06112-6
Keywords: L\'evy process, quadratic variation process, self-normalised process, Cram\'er bound, domain of attraction of the normal distribution for small times, law of the iterated logarithm for small times, Feller stochastic compactness classes
Received by editor(s): February 18, 2012
Received by editor(s) in revised form: January 23, 2013
Published electronically: October 9, 2014
Additional Notes: The first author’s research was partially supported by ARC Grant DP0988483
The second author’s research was partially supported by ARC Grant DP1092502
The third author’s research was partially supported by an NSF grant
Article copyright: © Copyright 2014 American Mathematical Society

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