Convergence of time series processes to continuous time limits

Abstract

Weak convergence of time series processes, as the length of the discrete time interval between observations tends to 0, has been studied extensively over the last decade. Existing results are obtained when the driving noise of time series processes are sequences of i.i.d random variables and there is no delay in the noise component of the limiting process.

The key focus of this thesis is to provide a unified theoretical framework under which the following remaining significant questions can be answered. * How can we obtain conditions for weak convergence of time series processes driven by a sequence of non i.i.d random variables? * How can we obtain conditions for weak convergence of time series processes to a limiting process having delays in both autoregressive and noise components?

The thesis is divided into two parts. In the first part (Chapters 3, and 4), we obtain conditions for weak convergence of some stochastic processes in the Skorokhod topology, including a class of moving average processes, a class of solutions to linear stochastic differential equations (SDEs) with delays, and a class of solutions to bilinear SDEs with delays.

In the second part (Chapters 5, 6, and 7), we apply the results of the first part to obtain conditions for weak convergence, as the length of the discrete time interval between observations tends to 0, of three classes of classical time series: moving average (MA), autoregressive moving average (ARMA), and generalized autoregressive conditional heteroskedastic (GARCH) processes. Our results are obtained under quite general conditions on the driving noise for these processes and considerably extend the assumptions of white noise typically imposed. Delays are allowed in both the autoregressive and noise components of the limiting processes of ARMA and GARCH processes. In particular, both COGARCH and Nelson diffusion types are permitted for GARCH limiting processes, and these are generalized to processes with delays. Sufficient conditions for strong existence and uniqueness of continuous time limits are given. Sufficient conditions for the existence of a stationary distribution for a continuous time limit of MA, ARMA processes, driven by Levy noise, are also obtained.