Studies in time series and random dynamics.

Asymptotic properties for various discrete-time stochastic processes are discussed. In particular, we investigate two important types of time series: linear processes and iterated function systems. Linear processes can exhibit long range dependence which has been observed in many fields, such as hydrology and economics. Iterated function systems contains some nonlinear time series models. Martingale techniques are extensively employed, which is shown to have advantage over the classical methodology of strong mixing and its associated blocking method. Martingale difference sequences are viewed as a natural generalization of independence. Using the martingale decomposition method, we are able to establish limit theories of those time series. Our results go beyond earlier ones in many aspects and solve several conjectures and open problems in the study of time series.PhDPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126845/2/3016988.pd