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

Lifshits, Mikhail; Linde, Werner

doi:10.1090/S0002-9947-04-03725-0

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

T. W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6 (1955), 170–176. MR 69229, DOI 10.1090/S0002-9939-1955-0069229-1
Eduard Belinsky and Werner Linde, Small ball probabilities of fractional Brownian sheets via fractional integration operators, J. Theoret. Probab. 15 (2002), no. 3, 589–612. MR 1922439, DOI 10.1023/A:1016263614257
Bernd Carl, On a characterization of operators from $l_{q}$ into a Banach space of type $p$ with some applications to eigenvalue problems, J. Functional Analysis 48 (1982), no. 3, 394–407. MR 678178, DOI 10.1016/0022-1236(82)90093-3
Bernd Carl, Ioanna Kyrezi, and Alain Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc. (2) 60 (1999), no. 3, 871–896. MR 1753820, DOI 10.1112/S0024610799008005
Bernd Carl and Irmtraud Stephani, Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR 1098497, DOI 10.1017/CBO9780511897467
Jakob Creutzig, Relations between classical, average, and probabilistic Kolmogorov widths, J. Complexity 18 (2002), no. 1, 287–303. MR 1895087, DOI 10.1006/jcom.2001.0622
S. Gengembre, Petites déviations pour les processus fractionnaires. Memoire de DEA, Université Lille-I, 2002.
James Kuelbs and Wenbo V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116 (1993), no. 1, 133–157. MR 1237989, DOI 10.1006/jfan.1993.1107
Wenbo V. Li, A Gaussian correlation inequality and its applications to small ball probabilities, Electron. Comm. Probab. 4 (1999), 111–118. MR 1741737, DOI 10.1214/ECP.v4-1012
Wenbo V. Li, Small ball probabilities for Gaussian Markov processes under the $L_p$-norm, Stochastic Process. Appl. 92 (2001), no. 1, 87–102. MR 1815180, DOI 10.1016/S0304-4149(00)00072-7
Wenbo V. Li and Werner Linde, Existence of small ball constants for fractional Brownian motions, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 11, 1329–1334 (English, with English and French summaries). MR 1649147, DOI 10.1016/S0764-4442(98)80189-4
Wenbo V. Li and Werner Linde, Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27 (1999), no. 3, 1556–1578. MR 1733160, DOI 10.1214/aop/1022677459
W. V. Li and Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., vol. 19, North-Holland, Amsterdam, 2001, pp. 533–597. MR 1861734, DOI 10.1016/S0169-7161(01)19019-X
M. A. Lifshits, Gaussian random functions, Mathematics and its Applications, vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736, DOI 10.1007/978-94-015-8474-6
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.
Mikhail A. Lifshits and Werner Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc. 157 (2002), no. 745, viii+87. MR 1895252, DOI 10.1090/memo/0745
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
Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437. MR 242239, DOI 10.1137/1010093
Ditlev Monrad and Holger Rootzén, Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Theory Related Fields 101 (1995), no. 2, 173–192. MR 1318191, DOI 10.1007/BF01375823
Alain Pajor and Nicole Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 4, 637–642. MR 845980, DOI 10.1090/S0002-9939-1986-0845980-8
A. Pietsch, Eigenvalues and $s$-numbers, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 43, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. MR 917067
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
G. Schechtman, Th. Schlumprecht, and J. Zinn, On the Gaussian measure of the intersection, Ann. Probab. 26 (1998), no. 1, 346–357. MR 1617052, DOI 10.1214/aop/1022855422
Qi Man Shao, A note on small ball probability of a Gaussian process with stationary increments, J. Theoret. Probab. 6 (1993), no. 3, 595–602. MR 1230348, DOI 10.1007/BF01066719
Nicole Tomczak-Jaegermann, Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 7, 299–301 (French, with English summary). MR 910364