Harris recurrent Markov chains and nonlinear monotone cointegrated models

In this paper, we study a nonlinear cointegration-type model of the form $Z_t = f_0(X_t) + W_t$ where $f_0$ is a monotone function and $X_t$ is a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator to locally estimate $f_0$, and under mild conditions, we show its strong consistency and obtain its rate of convergence. New results (of the Glivenko-Cantelli type) for localized null recurrent Markov chains are also proved.