The tail empirical process for long memory stochastic volatility models with leverage

We consider tail empirical processes of long memory stochastic volatility models with heavy tails and leverage. We study the limiting behaviour of the tail empirical process with both fixed and random levels. 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 mean estimators. The paper can be viewed as an extension of [21]; while the presence of leverage in the model creates additional theoretical problems, the limiting behaviour remains unchanged.