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)−W (t), 0 ≤ t ≤ 1 for 1/2 ≤ a < 1. We also prove a conjecture raised by Tanaka on the first moment of the limiting distribution of the least squares estimator (LSE) of a unit root process. The limiting random variable is a ratio of quadratic functionals of the m-times integrated Brownian motion. Its expectation can be found by using Karhunen-Loéve expansion and a property of the orthonormal eigenfunctions of the covariance function of the m-times integrated Brownian motion.