On tail behaviour and extremal values of some non-negative time series models

by Zhang, Zhiqiang

(Uncorrected OCR) Abstract of thesis entitled "On Tail Behaviour and Extremal Values of Some Non-negative Time Series Models" Submitted by ZHANG Zhiqiang For the Degree of Doctor of Philosophy at The University of Hong Kong in September 2002 This thesis mainly investigates the tail behaviour and extremal values for some non-negative time series models. First, some comments are made to the limit theory for stochastic difference equations and extremal values; one of the conditions in the limit theorem for multivariate stochastic difference equations is weakened. Next, several nonlinear time series models are considered. These time series models include two first order non-negative bilinear (BL(1)) models, one ith a uniform noise and another ith an exponential noise, an ARCH(2) model a GARCH(2,1) model and an ARCH(l) model driven by a mixture of normal noise (ARCH-MN(l) model). The stationarity, properties on moments and the tail behaviour and extremal values are studied, and a large amount of numerical results are provided. For the BL(1) model ith a uniform noise, we find its probability density function, and consider the tail behaviour hen the parameter takes a boundary value, hich complements the existing knoledge of tail behaviour of time se ries models. The numerically evaluated stationarity regions for ARCH(2) and GARCH(2,1) models are calculated for the first time. The comparison of the tail behaviour between the normal ARCH(l) and the ARCH-MN(l) models is also enlightening. The last part of this thesis investigates the distributions of non-Gaussian linear processes. We define a "closed" property for classes of distributions under the linear time series system and successfully find some closed classes of distributions, in hich the well knon class of normal distributions is included. We also find closed classes of infinitely divisible distributions, a-stable distributions as well as their mixtures. (25 words