The tail empirical process for some long memory sequences

TL;DR

This paper investigates the limiting behavior of tail empirical processes in long memory stochastic volatility models, revealing a dichotomy influenced by the Hurst parameter and tail index, with implications for tail index estimation.

Contribution

It introduces new theoretical results on tail empirical processes in long memory contexts and explores their non-Gaussian and degenerate limits, enhancing understanding of tail behavior.

Findings

Tail empirical process exhibits dichotomous behavior based on model parameters.

Tail empirical process with random levels remains unaffected by long memory.

New results for regularly varying distributions are established.

Abstract

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.