  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 flow 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 efficient estimators. The large-sample
distribution of such estimators is derived via empirical process theory for
cluster functionals. Their finite-sample performance is evaluated via Monte
Carlo simulations involving copula-based Markov models and solutions to
stochastic recurrence equations. The estimators are applied to stock price data
to study the absence or presence of symmetries in the succession of large gains
and losses.
