UC San Diego

In data rich environments we may sometimes deal with time series of infinite dimensional objects such as smooth curves, square-integrable functions or probability density functions. The so-called functional time series analysis considering such time series has been intensively developed by many authors. For stationary functional time series, we already have well-developed theoretical results. On the other hand, the literature on nonstationary functional time series is not rich enough yet; we only have a few recent papers that consider nonstationary functional time series.

This dissertation deals with functional cointegrated linear processes and develops a generalization of the Granger-Johansen representation theory in Hilbert spaces and Banach spaces. Given the scarcity of the existing literature on nonstationary functional time series, I believe that my results pave the way for the development of statistical methods involving functional time series exhibiting the random walk-type nonstationarity.

The first chapter concerns cointegrated linear processes in an arbitrary complex Hilbert space. We extend the notion of cointegration for time series taking values in such a Hilbert space, and provide generalized notions of I(0) and I(1) sequences. In the chapter we specifically show that the cointegrating space for an I(1) process may be sensibly defined as the kernel of the long run covariance operator of its first difference. Another main result of the chapter is a generalization of the Granger-Johansen representation theorem for I(1) autoregressive processes. We will observe that a geometric reformulation of the Johansen I(1) condition is useful to our Hilbert space setting, and it will be shown that a generalization of the Granger-Johansen theorem is derived based on this observation.

The second chapter is more focused on the Granger-Johansen theory in an arbitrary complex Hilbert space setting. We provide a generalization of the representation theorems for I(1) and I(2) autoregressive processes taking values in such a Hilbert space. A big difference from the previous chapter is that we rely on rigorous analytic operator-valued function theory. The most important input for our representation theory in the chapter is the so-called analytic Fredholm theorem, which will turn out to be useful. We will demonstrate this in detail.

In the last chapter, we will show that our representation theory based on the analytic Fredholm theorem can be extended to a Banach space setting. Specifically, we study the inversion of a holomorphic Fredholm operator-valued function in detail, and provide a closed-form expression of the inverse. Applying these results, we obtain our representation theory in an arbitrary complex separable Banach space. One meaningful aspect of this chapter is that we obtain a generalization of the Granger-Johansen theory without the help of rich geometric structure of a Hilbert space.