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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Small deviations of weighted fractional processes and average non–linear approximation

Authors: Mikhail A. Lifshits and Werner Linde
Journal: Trans. Amer. Math. Soc. 357 (2005), 2059-2079
MSC (2000): Primary 60G15; Secondary 47B06, 47B10, 47G10
Published electronically: December 9, 2004
MathSciNet review: 2115091
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Abstract: We investigate the small deviation problem for weighted fractional Brownian motions in $L_q$–norm, $1\le q\le \infty$. Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$. If $1/r:=H+1/q$, then our main result asserts \[ \lim _{\varepsilon \to 0} \varepsilon ^{1/H}\log \mathbb {P}\left ({\left \|{\rho B^H}\right \|_{L_q(0,\infty )}<\varepsilon } \right ) = -c(H,q)\cdot \left \|{\rho }\right \|_{L_r(0,\infty )}^{1/H}, \] provided the weight function $\rho$ satisfies a condition slightly stronger than the $r$–integrability. Thus we extend earlier results for Brownian motion, i.e. $H=1/2$, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non–linear approximation technique for Gaussian processes as well as sharp entropy estimates for $l_q$–sums of linear operators defined on a Hilbert space.

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Additional Information

Mikhail A. Lifshits
Affiliation: St. Petersburg State University, Postbox 104, 197372 St. Petersburg, Russia

Werner Linde
Affiliation: Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany

Keywords: Fractional Brownian motion, small deviations, fractional integration operator, entropy numbers, non–linear approximation
Received by editor(s): May 27, 2003
Received by editor(s) in revised form: December 18, 2003
Published electronically: December 9, 2004
Additional Notes: The authors were supported in part by DFG-RFBR Grant 99-01-04027 and by RFBR Grant 02-01-00265.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.