This thesis is about analysis of heavy-tailed time series. We discuss tail properties of real-world equity return series and investigate the possibility that a single tail index is shared by all return series of actively traded equities in a market. Conditions for this hypothesis to be true are identified.

We study the eigenvalues and eigenvectors of sample covariance and sample auto-covariance matrices of multivariate heavy-tailed time series, and particularly for time series with very high dimensions. Asymptotic approximations of the eigenvalues and eigenvectors of such matrices are found and expressed in terms of the parameters of the dependence structure, among others.

Furthermore, we study an importance sampling method for estimating rare-event probabilities of multivariate heavy-tailed time series generated by matrix recursion. We show that the proposed algorithm is efficient in the sense that its relative error remains bounded as the probability of interest tends to zero. We make use of exponential twisting of the transition kernel of an {\em Markov additive process}, and take advantage of asymptotic theories on products of positive random matrices.