Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

Eduard Belinsky () and Werner Linde ()
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Eduard Belinsky: University of the West Indies
Werner Linde: Friedrich-Schiller-Universität Jena

Journal of Theoretical Probability, 2002, vol. 15, issue 3, 589-612

Abstract: Abstract 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.

Keywords: Fractional integration; Kolmogorov numbers; entropy numbers; fractional Brownian motion; small ball behaviour (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (4)

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DOI: 10.1023/A:1016263614257

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