In this thesis we deal with statistical inference related to extreme value phenomena. Specifically, if X is a random vector with values in d-dimensional space, our goal is to estimate moments of ψ(X) for a suitably chosen function ψ when the magnitude of X is big. We employ the powerful tool of regular variation for random variables, random vectors and time series to formally define the limiting quantities of interests and construct the estimators. We focus on three statistical estimation problems: (i) multivariate tail estimation for regularly varying random vectors, (ii) extremogram estimation for regularly varying time series, (iii) estimation of the expected shortfall given an extreme component under a conditional extreme value model. We establish asymptotic normality of estimators for each of the estimation problems. The theoretical findings are supported by simulation studies and the estimation procedures are applied to some financial data.

We consider tail empirical processes for long memory stochastic volatility models with heavy tails and leverage. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic moment estimators. The limiting behaviour of these estimators is not affected by either long memory or leverage. Furthermore, we consider estimators of risk measures such as Value-at-Risk and Expected Shortfall. In these cases, the limiting behaviour is affected by long memory, but it is not affected by leverage. The theoretical results are illustrated by simulation studies.