| This work is a study of various long-range dependent heavy-tailed time series models, which are useful for modeling data in the fields of econometrics, communications, insurance, and finance. We examine self-normalized statistics for model parameters, such as the mean, in order provide theoretical backing for statistical inference in these models. These models have in common (in addition to strict stationarity) long-range dependence (the Joseph Effect) and heavy-tailed marginal distributions (the Noah Effect). Most of the models have a linear dependence structure (such as an infinite order moving average) and are strong mixing, i.e. asymptotically, the random variables are independent. In later chapters, generalizations to random fields are considered. My objective has been to conduct inference for model parameters---e.g. form asymptotically correct confidence intervals. Thus, the bulk of the results are weak limit theorems for statistics such as the sample mean.; Two theoretical/practical techniques occur as recurring themes in this work: self-normalization and subsampling. All of the statistics considered have "unknown" rates of convergence; this problem is solved by dividing or "normalizing" it by some other statistic which converges at that rate. A second difficulty is that the limit distributions generally are complicated functions of stable random variables and filter coefficients, and thus their quantiles are unknown; this is resolved by using subsampling methods to estimate the limit cumulative distribution function. Indeed, this work has the motive of providing a wealth of practical and useful probability models to which the already existing subsampling methods in [48] may be applied. |
