The tail empirical process for some long memory sequences

Rafal Kulik (1) , Philippe Soulier (2)

1 Department of Mathematics and Statistics [Ottawa]
2 MODAL'X - Modélisation aléatoire de Paris X

This paper describes limiting behaviour of tail empirical process associated with long memory stochastic volatility models. We show that such process has dichotomous behaviour, according to an interplay between a Hurst parameter and a tail index. In particular, the limit may be non-Gaussian and/or degenerate, indicating an influence of long memory. On the other hand, tail empirical process with random levels never suffers from long memory. This is very desirable from a practical point of view, since such the process may be used to construct Hill estimator of the tail index. To prove our results we need to establish several new results for regularly varying distribution functions, which may be of independent interest.

Mots clés: Tail empirical process; long memory processes; stochastic volatility

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Rafal Kulik, Philippe Soulier. The tail empirical process for some long memory sequences. Stochastic Processes and their Applications, 2011, 121 (1), pp.109-134. ⟨10.1016/j.spa.2010.09.001⟩. ⟨hal-00447960v2⟩