Karhunen–Loève expansion of Gaussian processes and Applications

Abstract

We study the Karhunen–Loève (KL) expansion of Gaussian processes and its application to (i) Inference of Unit distributions, (ii) Quantization of Gaussian processes, and (iii) Reduction of variance for estimating the expectation of Gaussian processes. This thesis has two original contributions. First, we propose a class of time-rescaled Brownian bridge, which is a novel deterministic time-changed Brownian bridge, and derive its closed-form KL expansion. We explore the relationship between its KL expansion and the associated empirical process, which leads to a new test statistic for the unit distributions. Also, we show that this test has greater power than existing tests. Second, we derive a novel connection between Malliavin derivatives and KL expansion. We prove the Malliavin differentiability of functionals of Hilbert space H-valued Gaussian process {X_t}_(t∈[0,T]) of the form φ(X_(t_1), . . . , X_(t_n)), where φ: H^n → R is Lipschitz. Integration by parts formula yields an expression for a covariance, which we then utilize for the estimation of E[φ(X_(t_1), . . . , X_(t_n))] using the control variate method of variance reduction.