Entropy Estimate for K-monotone Functions via Small Ball Probability of Integrated Brownian Motion

Small Ball Problem via Wavelets for Gaussian Processes

The recently developed theory of wavelets is used to estimate small ball probability for a class of Gaussian processes including d-dimensional fractional Brownian motion.

Hitting Estimates for Gaussian Random Fields

We provide estimates for the probability that a continuous, stationary Gaussian random eld hits a small ball. Our estimates are used to study the sample functions of (N, d) Brownian sheet and its corresponding integrated process.

Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Let Td : L2([0, 1] ) C([0, 1] ) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k(log k)d&1 . From this we derive that the small ball probabilities of the Brownian sheet on [0, 1] under the C([0, 1] )-norm can be estimated from below by exp(&C= |log =|), which improves the best known lower bounds considerably. We also get s...

Quadrati Fun tionals and Small Ball Probabilities for the m-fold Integrated Brownian Motion

Gaussian Processes : Karhunen - Loève Expansion , Small Ball

In this dissertation, we study the Karhunen-Loève (KL) expansion and the exact L small ball probability for Gaussian processes. The exact L small ball probability is connected to the Laplace transform of the Gaussian process via Sytaja Tauberian theorem. Using this technique, we solved the problem of finding the exact L small ball estimates for the Slepian process S(t) defined as S(t) = W (t+a)...