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.
"Nonparametric estimation in null recurrent time series." Ann. Statist. 29 (2) 372 - 416, April 2001. https://doi.org/10.1214/aos/1009210546