Statistics for Tail Processes of Markov Chains

At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process at the current state and on the ow of time, either forward or backward. Estimation of the tail process provides a nonparametric approach to analyze extreme values. A duality between the distributions of the forward and backward increments provides additional information that can be exploited in the construction of more e_cient estimators. The large-sample distribution of such estimators is derived via empirical process theory for cluster functionals. Their _nite-sample performance is evaluated via Monte Carlo simulations involving copula Markov models and solutions to stochastic recurrence equations. The estimators are applied to stock market data to study the absence or presence of symmetries in the succession of large losses and gains.