Limit theorems in functional data analysis with applications

This dissertation is concerned with functional data analysis. Functional data consists of a collection of curves or functions defined on an interval. These curves can be obtained by splitting a continuous time record such as temperature into daily or annual curves. Functional data is also obtained when an experimenter records a curve of data from each subject in a sample, e.g., a growth trajectory of an animal or plant. Several examples of different models for functional data are given. We use the method of principle component analysis to obtain the necessary regularization in each model. Functional principal component analysis is summarized as a natural extension of the traditional vector principal component analysis. The first functional model is concerned with inference based on the mean function of a functional time series. We develop and asymptotically justify a testing procedure for the equality of means in two functional samples exhibiting temporal dependence. As a second example, we consider a quadratic functional regression model in which a scalar response depends on a functional predictor. We develop a test of the significance of the nonlinear term in the model. The asymptotic behavior of our testing procedure is established. In the third model, we observe two sequences of curves which are connected via an integral operator. This model includes linear models as well as autoregressive models in Hilbert spaces. We develop a procedure to test the stability of the model. In the fourth model, we propose a functional version of the popular ARCH model. We establish conditions for the existence of a strictly stationary solution, derive weak dependence and moment conditions, show consistency of the estimators, and perform an empirical study demonstrating how our model matches with real data.