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

Author(s): Mikhail A. Lifshits; Werner Linde
Journal: Trans. Amer. Math. Soc. 357 (2005), 2059-2079.
MSC (2000): Primary 60G15; Secondary 47B06, 47B10, 47G10
Posted: December 9, 2004
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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

\begin{displaymath}\lim_{\varepsilon\to 0} \varepsilon^{1/H}\log \mathbb{P}\left... ...c(H,q)\cdot\left\Vert{\rho}\right\Vert _{L_r(0,\infty)}^{1/H}, \end{displaymath}

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:

1.
T. W. Anderson, The integral of symmetric unimodular functions over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 (1955), 170-176. MR 0069229 (16:1005a)

2.
E. S. Belinsky and W. Linde, Small ball probabilities of fractional Brownian sheets via fractional integration operators. J. Theor. Probab. 15 (2002), 589-612. MR 1922439 (2004d:60092)

3.
B. Carl, On a characterization of operators from $l_q$ into a Banach space of type $p$ with some applications to eigenvalue problems. J. Funct. Anal. 48 (1982), 394-407. MR 0678178 (84i:47033)

4.
B. Carl, I. Kyrezi and A. Pajor, Metric entropy of convex hulls in Banach spaces. J. London Math. Soc. 60 (1999), 871-896. MR 1753820 (2001c:46019)

5.
B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators. Cambridge Univ. Press, Cambridge 1990. MR 1098497 (92e:47002)

6.
J. Creutzig, Relations between classical, average, and probabilistic Kolmogorov widths. Complexity 18 (2002), 287-303. MR 1895087 (2002k:41041)

7.
S. Gengembre, Petites déviations pour les processus fractionnaires. Memoire de DEA, Université Lille-I, 2002.

8.
J. Kuelbs and W. V. Li, Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 (1993), 133-157. MR 1237989 (94j:60078)

9.
W. V. Li, A Gaussian correlation inequality and its applications to small ball probabilities. Electronic Commun. in Probability 12 (1999), 111-118. MR 1741737 (2001j:60074)

10.
W. V. Li, Small ball estimates for Gaussian Markov processes under $L_p$-norm. Stoch. Proc. Appl. 92 (2001), 87-102. MR 1815180 (2001m:60090)

11.
W. V. Li and W. Linde, Existence of small ball constants for fractional Brownian motions. C.R.A.S. Paris 326 (1998), Sér. I, 1329-1334. MR 1649147 (2000a:60077)

12.
W. V. Li and W. Linde, Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 (1999), 1556-1578. MR 1733160 (2001c:60059)

13.
W. V. Li and Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications. In: Shanbhag, D. N. (ed.) et al., Stochastic processes: Theory and methods. Amsterdam: North-Holland/Elsevier. Handb. Stat. 19 (2001), 533-597. MR 1861734

14.
M. A. Lifshits, Gaussian Random Functions Kluwer, Dordrecht, 1995. MR 1472736 (98k:60059)

15.
M. A. Lifshits, Asymptotic behavior of small ball probabilities. Probab. Theory and Math. Statist. Proc. VII International Vilnius Conference. Vilnius, VSP/TEV (1999), 453-468.

16.
M. A. Lifshits and W. Linde, Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion. Memoirs AMS 745 (2002), 1-87. MR 1895252 (2004g:47066)

17.
M. A. Lifshits and T. Simon, Small ball probabilities for stable Riemann-Liouville processes. Preprint of the University of Evry. (2003) arXiv: math.PR/0305092

18.
B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968), 422-437. MR 0242239 (39:3572)

19.
D. Monrad and H. Rootzén, Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Th. Rel. Fields 101 (1995), 173-192. MR 1318191 (96a:60032)

20.
A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite dimensional Banach spaces. Proc. Amer. Math. Soc. 97 (1985), 637-642. MR 0845980 (87i:46040)

21.
A. Pietsch, Eigenvalues and s-Numbers. Cambridge Univ. Press, Cambridge 1987. MR 0890520 (88j:47022b)

22.
G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge Univ. Press, Cambridge 1989. MR 1036275 (91d:52005)

23.
G. Schechtman, T. Schlumprecht and J. Zinn, On the Gaussian measure of the intersection. Ann. Probab. 26 (1998), 346-357. MR 1617052 (99c:60032)

24.
Q.-M. Shao, A note on small ball probability of Gaussian processes with stationary increments. J. Theoret. Probab. 6 (1993), 595-602. MR 1230348 (95b:60041)

25.
N. Tomczak-Jaegermann, Dualité des nombres d'entropie pour des opérateurs à valeurs dans un espace de Hilbert. C.R. Acad. Sci. Paris 305 (1987), 299-301. MR 0910364 (89c:47027)

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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

DOI: 10.1090/S0002-9947-04-03725-0
PII: S 0002-9947(04)03725-0
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
Posted: 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 of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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