  Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society:
Series B, 78(1), 31-51) recently introduced a stochastic model that allows for
heteroscedasticity of extremes. The model is extended to the situation where
the observations are serially dependent, which is crucial for many practical
applications. We prove a local limit theorem for a kernel estimator for the
scedasis function, and a functional limit theorem for an estimator for the
integrated scedasis function. We further prove consistency of a bootstrap
scheme that allows to test for the null hypothesis that the extremes are
homoscedastic. Finally, we propose an estimator for the extremal index
governing the dynamics of the extremes and prove its consistency. All results
are illustrated by Monte Carlo simulations. An important intermediate result
concerns the sequential tail empirical process under serial dependence.
