Mathematical Statistics : Time Series Analysis

Financial data have, among others, a particular feature: large values of such series cluster, we are concerned with estimation of clustering probabilities for univariate heavy tailed time series. We describe regular variation as a tool to model heavy tails. We summarize some results on the central limit theorem (CLT) and tightness of stochastic processes. These tools are needed to prove asymptotic normality of our estimator. We employ functional convergence of a bivariate tail empirical process,regular variation property and Lindeberg's CLT and the mixing property with geometric rates to conclude asymptotic normality of an estimator of the clustering probabilities. Theoretical results are illustrated by simulation studies.