We develop a nonparametric estimation theory in a nonstationary
environment, more precisely in the framework of null recurrent Markov chains.
An essential tool is the split chain, which makes it possible to decompose the
times series under consideration into independent and identical parts. A tail
condition on the distribution of the recurrence time is introduced. This
condition makes it possible to prove weak convergence results for sums of
functions of the process depending on a smoothing parameter. These limit
results are subsequently used to obtain consistency and asymptotic normality
for local density estimators and for estimators of the conditional mean and the
conditional variance. In contradistinction to the parametric case, the
convergence rate is slower than in the stationary case, and it is directly
linked to the tail behavior of the recurrence time. Applications to
econometric, and in particular to cointegration models, are indicated.