Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d≥2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.

[1]  B. Carl,et al.  Metric Entropy of Convex Hulls in Banach Spaces , 1999 .

[2]  B. S. Kašin,et al.  DIAMETERS OF SOME FINITE-DIMENSIONAL SETS AND CLASSES OF SMOOTH FUNCTIONS , 1977 .

[3]  M. Birman,et al.  PIECEWISE-POLYNOMIAL APPROXIMATIONS OF FUNCTIONS OF THE CLASSES $ W_{p}^{\alpha}$ , 1967 .

[4]  Werner Linde,et al.  Approximation, metric entropy and small ball estimates for Gaussian measures , 1999 .

[5]  V. N. Temli︠a︡kov.  Approximation of functions with bounded mixed derivative , 1989 .

[6]  Estimates for the Small Ball Probabilities of the Fractional Brownian Sheet , 2000 .

[7]  B. Carl.  Entropy numbers, s-numbers, and eigenvalue problems , 1981 .

[8]  A. Pietsch.  Eigenvalues and S-Numbers , 1987 .

[9]  A. S. ÜstünelÉcole.  Stochastic Analysis of the Fractional Brownian Motion , 1996 .

[10]  Thomas Kühn,et al.  Optimal series representation of fractional Brownian sheets , 2002 .

[11]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[12]  H. Rootzén,et al.  Small values of Gaussian processes and functional laws of the iterated logarithm , 1995 .

[13]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[14]  G. Lorentz.  Metric entropy and approximation , 1966 .

[15]  J. Kuelbs,et al.  Metric entropy and the small ball problem for Gaussian measures , 1993 .

[16]  A. Laptev.  Spectral asymptotic behavior of a class of integral operators , 1974 .

[17]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[18]  A note on small ball probability of a Gaussian process with stationary increments , 1993 .

[19]  V. Faber,et al.  Singular values of fractional integral operators: A unification of theorems of Hille, Tamarkin, and Chang , 1986 .

[20]  L. Werner.  Existence of small ball constants for fractional Brownian motions , 1998 .

[21]  Vladimir N. Temlyakov.  An Inequality for Trigonometric Polynomials and Its Application for Estimating the Entropy Numbers , 1995, J. Complex..

[22]  Thomas Kühn,et al.  Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet , 1999 .

[23]  Michel Talagrand,et al.  The Small Ball Problem for the Brownian Sheet , 1994 .

[24]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .