Gaussian processes: Karhunen-Loeve expansion, small ball estimates and applications in time series models
University of Delaware
In this dissertation, we study the Karhunen-Loève (KL) expansion and the exact L2 small ball probability for Gaussian processes. The exact L2 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 L2 small ball estimates for the Slepian process S (t ) defined as S (t ) = W (t + a ) - W (t ), t in [0,1], for a in [1/2,1). 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-Loève expansion and a property of the orthonormal eigenfunctions of the covariance function of the m -times integrated Brownian motion.