Small deviations of weighted fractional processes and average non–linear approximation
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- by Mikhail A. Lifshits and Werner Linde PDF
- Trans. Amer. Math. Soc. 357 (2005), 2059-2079 Request permission
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.References
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Additional Information
- Mikhail A. Lifshits
- Affiliation: St. Petersburg State University, Postbox 104, 197372 St. Petersburg, Russia
- Email: lifts@mail.rcom.ru
- Werner Linde
- Affiliation: Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany
- Email: lindew@minet.uni-jena.de
- 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.
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 2059-2079
- MSC (2000): Primary 60G15; Secondary 47B06, 47B10, 47G10
- DOI: https://doi.org/10.1090/S0002-9947-04-03725-0
- MathSciNet review: 2115091