Werner Linde

The Gaussian measure of shifted balls

SummaryLet μ be a centered Gaussian measure on a Hilbert spaceH and let

Small Deviation Probabilities of Sums of Independent Random Variables

B_R \subseteq H

Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes

be the centered ball of radiusR>0. Fora∈H and

Comparison Results for the Small Ball Behavior of Gaussian Random Variables

\mathop {\lim }\limits_{t{\mathbf{ }} \to {\mathbf{ }}\infty } {\mathbf{ }}R(t)/t< {\mathbf{ }}||a||

Small ball probabilities for integrals of weighted Brownian motion

, we give the exact asymptotics of μ(BR(t)+t·a) ast→∞. Also, upper and lower bounds are given when μ is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.

Localization of Majorizing Measures

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 Talagrands 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.