Journal of Theoretical Probability

, Volume 15, Issue 3, pp 589–612

Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

  • Eduard Belinsky
  • Werner Linde

DOI: 10.1023/A:1016263614257

Cite this article as:
Belinsky, E. & Linde, W. Journal of Theoretical Probability (2002) 15: 589. doi:10.1023/A:1016263614257


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.

Fractional integrationKolmogorov numbersentropy numbersfractional Brownian motionsmall ball behaviour

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Eduard Belinsky
    • 1
  • Werner Linde
    • 2
  1. 1.Department of Computer Science, Mathematics and PhysicsUniversity of the West IndiesBridgetownBarbados
  2. 2.Institut für StochastikFriedrich-Schiller-Universität JenaJenaGermany